Optomechanical Tooling

ABSTRACT

Optomechanical tools are disclosed. The optomechanical tools include a body of material having an entrance face, a rake face, a flank face, a rake side face, and a flank side face. The rake side face and the flank side face are connected to the entrance face. The rake side face is connected to the rake face. The flank side face is connected to the flank face. The rake face is connected to the flank face to define a curved cutting edge. The entrance face extends away from the flank side face to define a back-relief angle. The rake face extends away from the rake side face to define a rake angle. The entrance face is configured to direct a light beam toward one or more of the rake face, the flank face, the rake side face, the flank side face, and the curved cutting edge and through one or more of the rake face, the flank face, and the curved cutting edge, causing the light beam to refract onto the workpiece. Systems are also disclosed. Methods for transmitting a light beam through an optomechanical tool are also disclosed.

CROSS REFERENCE TO RELATED APPLICATIONS

This U.S. patent claims priority under 35 U.S.C. § 119(e) to U.S. Provisional Application 62/868,430, filed on Jun. 28, 2019, which is hereby incorporated by reference in its entirety.

TECHNICAL FIELD

This disclosure relates to optomechanical tooling, systems including optomechanical tooling and methodologies for utilizing systems including optomechanical tooling.

BACKGROUND

This section provides background information related to the present disclosure which is not necessarily prior art.

Light-assisted (e.g., laser-assisted) machining tools are known. While existing light-assisted machining tools perform adequately for their intended purpose, improvements to light-assisted machining tools are continuously being sought in order to advance the arts.

DESCRIPTION OF DRAWINGS

FIG. 1A is a side view of an exemplary optomechanical tool that receives a collimated light beam at a substantially linear, flat or planar light beam entrance face arranged at an obtuse back-relief angle.

FIG. 1B is a side view of an exemplary optomechanical tool that receives a collimated light beam at an inwardly-projecting axially cylindrical light beam entrance face arranged at an obtuse back-relief angle.

FIG. 1B′ is a perspective view of the optomechanical tool of FIG. 1B.

FIG. 1C is a side view of an exemplary optomechanical tool that receives a collimated light beam at an outwardly-projecting axially cylindrical light beam entrance face arranged at an obtuse back-relief angle.

FIG. 1C′ is a perspective view of the optomechanical tool of FIG. 1C.

FIG. 1D is a side view of an exemplary optomechanical tool that receives a collimated light beam at an inwardly-projecting laterally cylindrical light beam entrance face arranged at an obtuse back-relief angle.

FIG. 1D′ is a perspective view of the optomechanical tool of FIG. 1D.

FIG. 1E is a side view of an exemplary optomechanical tool that receives a collimated light beam at an outwardly-projecting laterally cylindrical light beam entrance face arranged at an obtuse back-relief angle.

FIG. 1E′ is a perspective view of the optomechanical tool of FIG. 1E.

FIG. 1F is a side view of an exemplary optomechanical tool that receives a collimated light beam at an inwardly-projecting spherical light beam entrance face arranged at an obtuse back-relief angle.

FIG. 1F′ is a perspective view of the optomechanical tool of FIG. 1F.

FIG. 1G is a side view of an exemplary optomechanical tool that receives a collimated light beam at an outwardly-projecting spherical light beam entrance face arranged at an obtuse back-relief angle.

FIG. 1G′ is a perspective view of the optomechanical tool of FIG. 1G.

FIG. 2A is a top view of the optomechanical tool of FIG. 1A.

FIG. 2B is a top view of the optomechanical tool of FIG. 1B.

FIG. 2C is a top view of the optomechanical tool of FIG. 1C.

FIG. 2D is a top view of the optomechanical tool of FIG. 1D.

FIG. 2E is a top view of the optomechanical tool of FIG. 1E.

FIG. 2F is a top view of the optomechanical tool of FIG. 1F.

FIG. 2G is a top view of the optomechanical tool of FIG. 1G.

FIG. 2H is a top view of an exemplary optomechanical tool including a light beam entrance face arranged at an obtuse back-relief angle defining one or more diffractive surface portions.

FIG. 2I is a top view of an exemplary optomechanical tool including a light beam entrance face arranged at an obtuse back-relief angle and a reflection-enhancing coating applied to one or more outer surfaces of the optomechanical tool.

FIG. 3A is a side view of an exemplary optomechanical tool that receives a collimated light beam at a substantially linear, flat or planar light beam entrance face arranged at an acute back-relief angle.

FIG. 3B is a side view of an exemplary optomechanical tool that receives a collimated light beam at an inwardly-projecting axially cylindrical light beam entrance face arranged at an acute back-relief angle.

FIG. 3B′ is a perspective view of the optomechanical tool of FIG. 3B.

FIG. 3C is a side view of an exemplary optomechanical tool that receives a collimated light beam at an outwardly-projecting axially cylindrical light beam entrance face arranged at an acute back-relief angle.

FIG. 3C′ is a perspective view of the optomechanical tool of FIG. 3C.

FIG. 3D is a side view of an exemplary optomechanical tool that receives a collimated light beam at an inwardly-projecting laterally cylindrical light beam entrance face arranged at an acute back-relief angle.

FIG. 3D′ is a perspective view of the optomechanical tool of FIG. 3D.

FIG. 3E is a side view of an exemplary optomechanical tool that receives a collimated light beam at an outwardly-projecting laterally cylindrical light beam entrance face arranged at an acute back-relief angle.

FIG. 3E′ is a perspective view of the optomechanical tool of FIG. 3E.

FIG. 3F is a side view of an exemplary optomechanical tool that receives a collimated light beam at an inwardly-projecting spherical light beam entrance face arranged at an acute back-relief angle.

FIG. 3F′ is a perspective view of the optomechanical tool of FIG. 3F.

FIG. 3G is a side view of an exemplary optomechanical tool that receives a collimated light beam at an outwardly-projecting spherical light beam entrance face arranged at an acute back-relief angle.

FIG. 3G′ is a perspective view of the optomechanical tool of FIG. 3G.

FIG. 4A is a top view of the optomechanical tool of FIG. 3A.

FIG. 4B is a top view of the optomechanical tool of FIG. 3B.

FIG. 4C is a top view of the optomechanical tool of FIG. 3C.

FIG. 4D is a top view of the optomechanical tool of FIG. 3D.

FIG. 4E is a top view of the optomechanical tool of FIG. 3E.

FIG. 4F is a top view of the optomechanical tool of FIG. 3F.

FIG. 4G is a top view of the optomechanical tool of FIG. 3G.

FIG. 4H is a top view of an exemplary optomechanical tool including a light beam entrance face arranged at an acute back-relief angle defining one or more diffractive surface portions.

FIG. 4I is a top view of an exemplary optomechanical tool including a light beam entrance face arranged at an acute back-relief angle and a reflection-enhancing coating applied to one or more outer surfaces of the optomechanical tool.

FIG. 5A is a side view of an exemplary optomechanical tool that receives a collimated light beam at a substantially linear, flat or planar light beam entrance face arranged at a perpendicular or right back-relief angle.

FIG. 5B is a side view of an exemplary optomechanical tool that receives a collimated light beam at an inwardly-projecting axially cylindrical light beam entrance face arranged at a perpendicular or right back-relief angle.

FIG. 5B′ is a perspective view of the optomechanical tool of FIG. 5B.

FIG. 5C is a side view of an exemplary optomechanical tool that receives a collimated light beam at an outwardly-projecting axially cylindrical light beam entrance face arranged at a perpendicular or right back-relief angle.

FIG. 5C′ is a perspective view of the optomechanical tool of FIG. δC.

FIG. 5D is a side view of an exemplary optomechanical tool that receives a collimated light beam at an inwardly-projecting laterally cylindrical light beam entrance face arranged at a perpendicular or right back-relief angle.

FIG. 5D′ is a perspective view of the optomechanical tool of FIG. 5D.

FIG. 5D _(C) is a side view of an exemplary optomechanical tool that receives a converging light beam at an inwardly-projecting laterally cylindrical light beam entrance face arranged at a perpendicular or right back-relief angle.

FIG. 5D _(D) is a side view of an exemplary optomechanical tool that receives a diverging light beam at an inwardly-projecting laterally cylindrical light beam entrance face arranged at a perpendicular or right back-relief angle.

FIG. 5E is a side view of an exemplary optomechanical tool that receives a collimated light beam at an outwardly-projecting laterally cylindrical light beam entrance face arranged at a perpendicular or right back-relief angle.

FIG. 5E′ is a perspective view of the optomechanical tool of FIG. 5E.

FIG. 5E _(C) is a side view of an exemplary optomechanical tool that receives a converging light beam at an outwardly-projecting laterally cylindrical light beam entrance face arranged at a perpendicular or right back-relief angle.

FIG. 5E _(D) is a side view of an exemplary optomechanical tool that receives a diverging light beam at an outwardly-projecting laterally cylindrical light beam entrance face arranged at a perpendicular or right back-relief angle.

FIG. 5F is a side view of an exemplary optomechanical tool that receives a collimated light beam at an inwardly-projecting spherical light beam entrance face arranged at a perpendicular or right back-relief angle.

FIG. 5F′ is a perspective view of the optomechanical tool of FIG. 5F.

FIG. 5G is a side view of an exemplary optomechanical tool that receives a collimated light beam at an outwardly-projecting spherical light beam entrance face arranged at a perpendicular or right back-relief angle.

FIG. 5G′ is a perspective view of the optomechanical tool of FIG. 5G.

FIG. 6A is a top view of the optomechanical tool of FIG. 5A.

FIG. 6B is a top view of the optomechanical tool of FIG. 5B.

FIG. 6C is a top view of the optomechanical tool of FIG. 5C.

FIG. 6D is a top view of the optomechanical tool of FIG. 5D.

FIG. 6E is a top view of the optomechanical tool of FIG. 5E.

FIG. 6F is a top view of the optomechanical tool of FIG. 5F.

FIG. 6G is a top view of the optomechanical tool of FIG. 5G.

FIG. 6H is a top view of an exemplary optomechanical tool including a light beam entrance face arranged at a perpendicular or right back-relief angle defining one or more diffractive surface portions.

FIG. 6I is a top view of an exemplary optomechanical tool including a light so beam entrance face arranged at a perpendicular or right back-relief angle and a reflection-enhancing coating applied to one or more outer surfaces of the optomechanical tool.

FIG. 7A is a side view of an exemplary optomechanical tool that receives a collimated light beam at a substantially linear, flat or planar functional entrance face segment of a light beam entrance face arranged at an acute back-relief angle.

FIG. 7B is a side view of an exemplary optomechanical tool that receives a collimated light beam at an inwardly-projecting axially cylindrical functional entrance face segment of a light beam entrance face arranged at an acute back-relief angle.

FIG. 7B′ is a perspective view of the optomechanical tool of FIG. 7B.

FIG. 7C is a side view of an exemplary optomechanical tool that receives a collimated light beam at an outwardly-projecting axially cylindrical functional entrance face segment of a light beam entrance face arranged at an acute back-relief angle.

FIG. 7C′ is a perspective view of the optomechanical tool of FIG. 7C.

FIG. 7D is a side view of an exemplary optomechanical tool that receives a collimated light beam at an inwardly-projecting laterally cylindrical functional entrance face segment of a light beam entrance face arranged at an acute back-relief angle.

FIG. 7D′ is a perspective view of the optomechanical tool of FIG. 7D.

FIG. 7E is a side view of an exemplary optomechanical tool that receives a collimated light beam at an outwardly-projecting laterally cylindrical functional entrance face segment of a light beam entrance face arranged at an acute back-relief angle.

FIG. 7E′ is a perspective view of the optomechanical tool of FIG. 7E.

FIG. 7F is a side view of an exemplary optomechanical tool that receives a collimated light beam at an inwardly-projecting spherical functional entrance face segment of a light beam entrance face arranged at an acute back-relief angle.

FIG. 7F′ is a perspective view of the optomechanical tool of FIG. 7F.

FIG. 7G is a side view of an exemplary optomechanical tool that receives a collimated light beam at an outwardly-projecting spherical functional entrance face segment of a light beam entrance face arranged at an acute back-relief angle.

FIG. 7G′ is a perspective view of the optomechanical tool of FIG. 7G.

FIG. 8A is a top view of the optomechanical tool of FIG. 7A.

FIG. 8B is a top view of the optomechanical tool of FIG. 7B.

FIG. 8C is a top view of the optomechanical tool of FIG. 7C.

FIG. 8D is a top view of the optomechanical tool of FIG. 7D.

FIG. 8E is a top view of the optomechanical tool of FIG. 7E.

FIG. 8F is a top view of the optomechanical tool of FIG. 7F.

FIG. 8G is a top view of the optomechanical tool of FIG. 7G.

FIG. 8H is a top view of an exemplary optomechanical tool including a functional entrance face segment of a light beam entrance face arranged at an acute back-relief angle defining one or more diffractive surface portions.

FIG. 8I is a top view of an exemplary optomechanical tool including a functional entrance face segment of a laser beam entrance face arranged at an acute back-relief angle and a reflection-enhancing coating applied to one or more outer surfaces of the optomechanical tool.

FIG. 9 is a side view of an exemplary optomechanical tool that receives a collimated light beam at a substantially linear, flat or planar light beam entrance face arranged at an obtuse back-relief angle and a secondary clearance face.

FIG. 10 is a top view of the optomechanical tool of FIG. 9.

FIG. 11A is a side view of an exemplary optomechanical tool that receives a collimated light beam at a substantially linear, flat or planar light beam entrance face arranged at an acute back-relief angle and a secondary clearance face.

FIG. 11B is another side view of an exemplary optomechanical tool that receives a collimated light beam at a substantially linear, flat or planar light beam entrance face arranged at an acute back-relief angle and a secondary clearance face.

FIG. 12A is atop view of the optomechanical tool of FIG. 11A.

FIG. 12B is a top view of the optomechanical tool of FIG. 11B.

FIG. 13 is a side view of an exemplary optomechanical tool that receives a collimated light beam at a substantially linear, flat or planar light beam entrance face arranged at a perpendicular or right back-relief angle and a secondary clearance face.

FIG. 14 is a top view of the optomechanical tool of FIG. 13.

FIG. 15 is a side view of an exemplary optomechanical tool that receives a collimated light beam at a substantially linear, flat or planar light beam entrance face arranged at a perpendicular or right back-relief angle, a secondary clearance face, and no rake side face.

FIG. 16 is a top view of the optomechanical tool of FIG. 15.

FIG. 17 is a side view of an exemplary optomechanical tool that receives a collimated light beam at an inwardly-projecting wedge shape light beam entrance face arranged at a perpendicular or right back-relief angle.

FIG. 17′ is a perspective view of the optomechanical tool of FIG. 17.

FIG. 18 is a top view of the optomechanical tool of FIG. 17.

FIG. 19 is a side view of an exemplary optomechanical tool that receives a collimated light beam at an outwardly-projecting wedge shape light beam entrance face arranged at a perpendicular or right back-relief angle.

FIG. 19′ is a perspective view of the optomechanical tool of FIG. 19.

FIG. 20 is a top view of the optomechanical tool of FIG. 19.

FIG. 21 is a side view of an exemplary optomechanical tool that receives a collimated light beam at a light beam entrance face including one or more diffractive surface portions arranged at a perpendicular or right back-relief angle.

FIG. 22 is a side view of an exemplary optomechanical tool that receives a collimated light beam at a non-movable or fixed optical lens arranged within an optical lens recess formed by a light beam entrance face of the optomechanical tool.

FIGS. 23A-23C are top views of an exemplary optomechanical tool that receives a collimated light beam at movable optical lens arranged near a light beam entrance face of the optomechanical tool.

FIGS. 24A-24C are top views of an exemplary optomechanical tool that receives a collimated light beam at an optical prism system arranged near a light beam entrance face of the optomechanical tool.

FIG. 25 is a side view of an exemplary optomechanical tool that receives a collimated light beam at an outwardly-projecting light beam entrance face arranged at a perpendicular or right back-relief angle.

FIG. 25′ is a perspective view of the optomechanical tool of FIG. 25.

FIG. 26 is a top view of the optomechanical tool of FIG. 25.

FIG. 26′ is an enlarged view according to line 26′ of FIG. 26.

FIG. 27 is a plan view of an exemplary optomechanical tool.

FIG. 28 is a side view of the optomechanical tool of FIG. 27 transmitting a light beam.

FIG. 29 is a view of the optomechanical tool of engaging a workpiece while transmitting the light beam.

FIG. 30A is a side view of the optomechanical tool of FIG. 27 arranged relative a workpiece having a highest compression region extending along at least a rake face of the optomechanical tool and a lowest tensile region extending across a flank face of the optomechanical tool.

FIG. 30B is a side view of the optomechanical tool of FIG. 27 arranged relative a workpiece having a high compression region extending along at least a rake face of the optomechanical tool and a low tensile region extending across a flank face of the optomechanical tool.

FIG. 30C is a side view of the optomechanical tool of FIG. 27 arranged relative a workpiece having a medium compression region extending along at least a rake face of the optomechanical tool and a medium tensile region extending across a flank face of the optomechanical tool.

FIG. 30D is a side view of the optomechanical tool of FIG. 27 arranged relative a workpiece having a low compression region extending along at least a rake face of the optomechanical tool and a high tensile region extending across a flank face of the optomechanical tool.

FIG. 30E is a side view of the optomechanical tool of FIG. 27 arranged relative a workpiece having a lowest compression region extending along at least a rake face of the optomechanical tool and a highest tensile region extending across a flank face of the optomechanical tool.

FIG. 31A is a perspective view of an exemplary light beam.

FIG. 31A is an end view of the light beam of FIG. 31A.

FIG. 32 is a schematic view of an exemplary computing device.

Like reference symbols in the various drawings indicate like elements.

DETAILED DESCRIPTION

Aspects of the present disclosure are directed to systems including a light (e.g., laser) generator and optomechanical tooling. The optomechanical tooling may so machine a workpiece defined by a material (e.g., ceramics, semiconductors, optical crystals, glass, metal alloys, plastics, composites, bone, teeth, and the like) that minimizes tooling forces while improving surface finish, aesthetics, form repeatability, and overall machinability of the workpiece.

Although the following disclosure describes a variety of configurations of optomechanical tooling, which may be alternatively referred to as, for example, a “laser-transmitting machining tool,” that receives, for example, a laser light beam, the optomechanical tooling/laser-transmitting machining tool(s) is/are not limited to receiving a laser light beam; accordingly, the optomechanical tooling/laser-transmitting machining tool(s) may receive any type of light such as, for example, visible light (having a wavelength between 380 nm and 780 nm), infrared light (having a wavelength longer than 780 nm), ultraviolet light (having a wavelength shorter than 380 nm), laser light (having a wavelength between 150 nm and 1100 nm), and the like. Accordingly, although the term “laser” is utilized in the following disclosure in association with the optomechanical tooling/laser-transmitting machining tool(s), the term “laser” is utilized as an exemplary form of light, and, as such, the optomechanical tooling/laser-transmitting machining tool(s) described in the present disclosure is/are not limited to receiving one type of light beam, such as, for example a laser beam.

Other aspects of the present disclosure include methodologies for utilizing the systems including a laser-transmitting machining tool for machining the workpiece. In an example, after directly engaging the workpiece with the laser-transmitting machining tool, the laser-transmitting machining tool transmits laser radiation from the laser generator to the portions of the workpiece for the purpose of weakening the bonds of the workpiece and therefor softening the workpiece in order to ultimately plastically deform and/or thermally soften the workpiece.

Yet other aspects of the present disclosure include systems that may be interfaced with any of the laser-transmitting machining tool. In some instances, exemplary systems may be utilized for characterizing and qualifying the laser beam transmitted by the laser-transmitting tool. In other examples, systems may determine the laser power reflected, transmitted, and absorbed by the laser-transmitting machining tool while machining a workpiece. Even further, in other examples, exemplary the systems may also characterize and qualify the laser beam transmitted by the laser-transmitting tool when the laser-transmitting tool is not in contact with the workpiece. In yet other examples, exemplary systems may precisely measure the size, shape, and position of the laser beam transmitted by the laser-transmitting machining tool. In further examples, exemplary systems may further compare the laser size, shape, and position with a fiducial.

Prior to describing configurations of exemplary laser-transmitting machining tools 10 a-10 o at FIGS. 1-26′, reference is made to FIGS. 27-30E that are directed to exemplary laser-transmitting machining tools 10. Referring initially to FIG. 27, an exemplary laser-transmitting machining tool 10 defines a plurality of surfaces or faces 12-20. The surface 12 of the plurality of surfaces or faces 12-20 may be referred to as a laser beam entrance face. The surface 14 of the plurality of surfaces or faces 12-20 may be referred to as a rake face. The surface 16 of the plurality of surfaces or faces 12-20 may be referred to as a flank face or clearance face. The surface 18 of the plurality of surfaces or faces 12-20 may be referred to as a first side face or a rake side face. The surface 20 of the plurality of surfaces or faces 12-20 may be referred to as a second side face or a flank side face.

A first end 18, of the first side face 18 extends away from a first end 12 ₁ of the laser beam entrance face 12. A first end 20 ₁ of the second side face 20 extends away from a second end 12 ₂ of the laser beam entrance face 12.

A first end 14 ₁ of the rake face 14 extends away from a second end 18 ₂ of the first side face 18. A first end 16 ₁ of the flank face 16 extends away from a second end 20 ₂ of the second side face 20. A second end 14 ₂ of the rake face 14 is joined to a second end 16 ₂ of the flank face 16 to define a cutting edge 22. Furthermore, the first end 14 ₁ of the rake face 14 extends away from the second end 18 ₂ of the first side face 18 at a rake angle θ₁₄, and the first end 16 ₁ of the flank face 16 extends away from the second end 20 ₂ of the second side face 20 at a flank angle or clearance angle θ₁₆. The angle θ₁₄ defined by the rake face 14 and the first side face 18 may be referred to as a rake angle. The angle θ₁₆ defined by the flank face 16 and the second side face 20 may be referred to as a flank angle or clearance angle. As will be described in greater detail with respect to FIGS. 30A-30E, the rake angle θ₁₄ and the flank angle θ₁₆ are described in the context of the laser-transmitting machining tool 10 itself and not with respect to a surrounding environment relative the laser-transmitting machining tool 10 such as, for example, how the laser-transmitting machining tool 10 is positioned relative to a workpiece (see, e.g., W in FIG. 29).

One or more surfaces (see, e.g., laser beam entrance face 12) of the plurality of surfaces or faces 12-20 may define a laser beam entrance end 24 of the laser-transmitting machining tool 10. Further, one or more surfaces (see, e.g., rake face 14 and flank face 16) of the plurality of surfaces or faces 12-20 may define a laser beam exit end 26 of the laser-transmitting machining tool 10.

Furthermore, one or more surfaces (see, e.g. rake face 14 and first side face 18) of the plurality of surfaces or faces 12-20 may define a first side 28 of the laser-transmitting machining tool 10. Furthermore, one or more surfaces (see, e.g. laser beam entrance face 12, flank face16, and second side face 20) of the plurality of surfaces or faces 12-20 may define a second side 30 of the laser-transmitting machining tool 10.

The laser-transmitting machining tool 10 defines a tool length l. In an example, the tool length l is bound by the first end 18 ₁ of the first side face 18 and the cutting edge 22.

Furthermore, the laser-transmitting machining tool 10 may also include an anti-reflective coating 32 applied to at least one of the plurality of surfaces or faces 12-20 of the laser-transmitting machining tool 10. In an example, the anti-reflective coating 32 may be applied to the laser beam entrance face 12

Inclusion of a heat-activated/laser-activated cutting fluid/slurry/etchant upon one or both of the cutting edge 22, rake face 14, and flank face 16, permits the laser-transmitting machining tool 10 to chemically react in response to being subjected to heat or exposure of a laser beam L when the laser beam L exits the exit end 26 of the laser-transmitting machining tool 10. After reaction of the heat-activated/laser-activated cutting fluid/slurry/etchant and arranging the laser-transmitting machining tool 10 adjacent the workpiece W, a more ductile regime of material removal is promoted. Accordingly, the removal rate of material from the workpiece W may be increased while also using less tooling forces imparted from the laser-transmitting machining tool 10.

As seen in FIG. 27, the laser beam L is transmitted through the laser-transmitting machining tool 10. The laser beam L is directed from a laser generator toward the laser beam entrance end 24 of the laser-transmitting machining tool 10. The laser beam L enters the laser-transmitting machining tool 10 at the laser beam entrance face 12 at a relief angle θ_(i) relative to a line R that is normal to the laser beam entrance face 12. The laser beam L is then refracted within the laser-transmitting machining tool 10 at an angle θ_(r) and travels along the length l of the laser-transmitting machining tool 10 from the laser beam entrance end 24 of the laser-transmitting machining tool 10 to the laser beam exit end 26 of the laser-transmitting machining tool 10.

With reference to FIGS. 31A and 31B, the laser beam L defines a laser beam diameter Φ. The laser beam diameter Φ may further define: a central ray Φ_(A) extending along a central axis L_(A)-L_(A) (see, e.g., FIG. 31A) of the laser beam L; a first circumferential array of rays Φ_(R1) arranged at a first radial distance away from the central axis L_(A)-L_(A) of the laser beam L; and at least one second circumferential array of rays Φ_(R2) arranged at a second radial distance away from the central axis L_(A)-L_(A) of the laser beam L whereby the second radial distance is greater than the first radial distance.

With reference to FIG. 28, according to the refraction principles of light, the laser beam L will undergo another refraction when exiting the laser-transmitting machining tool 10 provided that the laser beam L strikes the laser beam flank face 16 with less than a critical angle (see, e.g., θ_(c) in Equation 3) when going from a first medium (e.g., a diamond material) of a higher refractive index m to a second medium (e.g., air) of a lower refractive index n₁. Assuming n₁=1 for air, the governing relationship is given by:

$\begin{matrix} {{\sin \; \theta_{C}} = \frac{1}{n_{2}}} & (1) \end{matrix}$

The general governing relationship is given by:

$\begin{matrix} {{\sin \; \theta_{C}} = \frac{n_{1}}{n_{2}}} & (2) \end{matrix}$

Accordingly, the critical angle θ_(C) is governed by the following equation:

$\begin{matrix} {\theta_{C} = {\sin^{- 1}\left( \frac{n_{1}}{n_{2}} \right)}} & (3) \end{matrix}$

In an example, for a laser beam L transitioning from diamond to air, a diamond material may have a critical angle of 24.4°; any incident laser beam L striking a surface greater than this angle will reflect internally in the diamond. In an example, FIG. 28 illustrates exemplary reflected rays Φ_(R1), Φ_(R2) exiting the laser beam exit end 26 that are directed from the laser beam entrance face 12 to the rake face 14.

With reference to FIG. 29, at least a portion of the laser beam exit end 26 of the laser-transmitting machining tool 10 contacts, is disposed adjacent, or is immersed into a workpiece W during the machining process. The material defining the workpiece W may include but not limited to ceramics, semiconductors, optical crystals, glass, metal alloys, plastics, composites, bone, teeth, and the like. Arranging the laser-transmitting machining tool 10 adjacent or immersing the laser-transmitting machining tool 10 into a volume of the workpiece W allow the rays Φ_(A), Φ_(R1), Φ_(R2) of laser beam L to be transmitted into and absorbed by selected portions of the workpiece W as the index of refraction n of the workpiece W is higher than the index of refraction n₁ of air, which results in an increase of the critical angle for internal reflection.

In an example, an exemplary laser-transmitting machining tool 10 composed of silicon may be defined by an index of refraction u equal to 3.4 such that no limitation for internal reflection exists as the workpiece W being machined has a higher index of refraction m compared to the index of refraction n of an exemplary laser-transmitting machining tool 10 composed of a diamond. The rays Φ_(A), Φ_(R1), Φ_(R2) of a laser beam L will enter the immersed area of a workpiece W, allowing the laser beam L to treat a selected region of the workpiece W undergoing compressive stresses effectively. Accordingly, as seen in FIG. 29, the rays Φ_(R1), Φ_(R2) of the laser beam L exiting the rake face 14 are allowed to propagate into the workpiece W of similar or higher index of refraction whereas the rays Φ_(R1), Φ_(R2) of the laser beam L exiting the flank face 16 represent a portion of the laser beam L affecting the workpiece W that had already been machined by the flank face 16 and the cutting edge 22 (i.e., the flank face 16 anneals the workpiece W so as the flank face 16 contacts the workpiece W).

As seen in FIG. 29, the central ray Φ_(A) of the laser beam L is focused on and exits the cutting edge 22 of the laser-beam exit end 26 of the laser-transmitting machining tool 10. As explained above, in addition to the laser beam L exiting the cutting edge 22 of the laser beam exit end 26 of the laser-transmitting machining tool 10, the laser beam L also exits one or both of the rake face 14 of the laser beam exit end 26 of the laser-transmitting machining tool 10 and the flank face 16 of the laser beam exit end 26 of the laser-transmitting machining tool 10. In an example, some of the first and second circumferential array of rays Φ_(R1), Φ_(R2) may exit the rake face 14 and some of the first and second circumferential army of rays Φ_(R1), Φ_(R2) may exit the flank face 16.

With continued reference to FIG. 29, the laser beam exit end 26 of the laser-transmitting machining tool 10 may be disposed adjacent a workpiece W that is plastically deformed and/or thermally softened by the laser-transmitting machining tool 10. The workpiece W may generally define a compressive region W_(C) and a tensile region W_(T).

In some instances, the compression region W_(C) of the workpiece W may generally extend across the rake face 14 and a portion of the flank face 16 near the second end 16 ₂ of the flank face 16 (i.e., the compression region W_(C) of the workpiece W extends across the cutting edge 22 of the laser-transmitting machining tool 10). In some examples, the tensile region W_(T) of the workpiece W may generally extend across the flank face 16 of the laser-transmitting machining tool 10 near the second end 16 ₂ of the flank face 16 without extending across the cutting edge 22 of the laser-transmitting machining tool 10. In other examples, the tensile region W_(T) of the workpiece W may generally extend from the flank face 16 and across the cutting edge 22 such that the tensile region WT of the workpiece W extends slightly across the rake face 14 of the laser-transmitting machining tool 10 near the second end 14 ₂ of the rake face 14. In some instances, the tensile region W_(T) may extend slightly across the rake face 14, and, in such instances, the tensile region W_(T) extending slightly across the rake face 14 is not limited to the geometry of the laser-transmitting tool 10, the material of the workpiece W, processing parameters, and the like.

Referring to FIGS. 30A-30E, one, or both of the rake angle θ₁₄ and the flank angle θ₁₆ may correspond to one or more qualities of a material of a workpiece W that is to be machined by the laser-transmitting machining tool 10. In an example, the rake angle θ₁₄ may range between approximately about 91° and 195° the flank angle θ₁₆ may range between approximately about 93° and 120°. The one or more qualities of the material of a workpiece W may be related to different levels of a compressive force imparted from the laser-transmitting machining tool 10 to the compression region W_(C) of the workpiece W and a tensile force imparted from the laser-transmitting machining tool 10 to the tensile region W_(T) of the workpiece W.

In an example, the rake angle θ₁₄ of FIG. 30A may be referred to as a highly negative rake angle and may be greater than 90° less than about 135°. The rake angle θ₁₄ of FIG. 30B may be referred to as a midrange negative rake angle, which may be greater than the highly negative rake angle θ₁₄ of FIG. 30A; in an example, the midrange negative rake angle θ₁₄ may be greater than about 136° and less than about 165°. The rake angle θ₁₄ of FIG. 30C may be referred to as a low-range negative rake angle, which may be greater than the midrange negative rake angle θ₁₄ of FIG. 30B; in an example, the low-range negative rake angle θ₁₄ may be greater than about 166° and less than about 179°. The rake angle θ₁₄ of FIG. 30D may be referred to as a zero rake angle, which is greater than the low-range negative rake angle θ₁₄ of FIG. 30C; in an example, the zero rake angle may be approximately equal to 180°. The rake angle θ₁₄ of FIG. 30E may be referred to as a positive rake angle, which may be greater than the zero rake angle θ₁₄ of FIG. 30D; in an example, the positive rake angle θ₁₄ may be greater than about 181° and less than about 210°. With reference to Table 1, exemplary materials and corresponding exemplary ranges of rake angles θ₁₄ are shown below.

TABLE 1 Material Of The Workpiece W Rake Angle θ₁₄ Range Silicon About 135° to About 155° Zinc Selenide About 145° to About 165° Zinc Sulfide About 145° to About 165° Calcium Fluoride About 145° to About 165° Tungsten Carbide About 145° to About 180° Aluminum About 175° to About 190° Steel or Stainless Steel About 91° to About 135° Germanium About 91° to About 135° Glass About 91° to About 135° Sapphire About 91° to About 135° Spinel About 135° to About 165° Barium Fluoride About 135° to About 165°

In an example, the highly negative rake angle θ₁₄ of FIG. 30A of the midrange negative rake angle θ₁₄ of FIG. 30B may be a preferable configuration of the laser-transmitting machining tool 10 when the material defining the workpiece W is, for example, a ceramic or optical crystal material that is stronger in compression with respect to tension (i.e., the forces involved in machining the compression region W_(C) are comparatively greater the tensile region W_(T)). In addition to design consideration of one or both of the rake angle θ₁₄ and the flank angle θ₁₆, the laser beam L radiated from the laser beam exit end 26 of the laser-transmitting machining tool 10 may also be selectively adjusted in order to compensate for known compressive and tensile qualities of the workpiece W.

In another example, the highly negative rake angle θ14 may be an angle ranging between about 135° and about 155° for machining a workpiece W (see, e.g., any of FIGS. 29, and 30A-30E) derived from a silicon material with a laser beam L focused on the cutting edge 22 but also biased toward the rake face 14 in order to promote plastic deformation, thermal softening and ductile removal of material in the compression region W_(C) of the workpiece W. Alternatively, if desired, the laser beam B may be focused on the cutting edge 22 but also biased toward the flank face 16 in order to minimize sub-surface damage to the tensile region WT of the workpiece W and promote an annealing or “healing” effect of the workpiece W. Accordingly, the act of biasing the laser beam L toward the rake face 14 promotes a more ductile cutting regime, allowing an increased cutting depth. Thus, the removal rate of material from the workpiece W may be increased while preserving the integrity of the laser-transmitting machining tool 10. Furthermore, post-processing (e.g., polishing) of the workpiece W may be minimized or eliminated if the laser beam L is biased toward the flank face 16.

In yet another example with reference to FIG. 30D, a zero rake angle θ₁₄ may be selected for machining a workpiece W (see, e.g., any of FIGS. 29, and 30A-30E) derived from a metal or metal composition due to the fact that most metals (such as, e.g., aluminum) are stronger in tensile with respect to compression; therefore, positive rake angles θ₁₄ (see, e.g., FIG. 30E) or rake angles θ₁₄ close to 180° (see, e.g. FIG. 30C) may be utilized for machining metallic or polymeric materials. Composite materials, however, are of many types and therefore material composition will control tool geometry. Accordingly, in order to promote the machinability in the tensile region for a material having a strong tensile quality, the laser beam L may be focused on the cutting 10 edge 22 but also biased toward the flank face 16 or rake face 14 in order to promote plastic deformation, thermal softening, and removal of material in the tensile region WT of the workpiece W.

With reference to FIG. 27, the act of biasing of the laser beam to one of the rake face 14 and the flank face 16 of the laser-beam exit end 26 of the laser-transmitting machining tool 10 is described as follows. In an example, the laser-transmitting machining tool 10 of FIG. 27 may be defined by a midrange negative rake angle θ₁₄, and based on Snell's law, the minimum relief angle θ_(i) can be calculated given a known length l of the laser-transmitting machining tool 10 and a desired location (see, e.g., horizontal line a) below the cutting edge 22.

When light (i.e., the laser beam L) enters a medium of a higher refractive index n₂ (i.e., the medium defined by the laser-transmitting machining tool 10), the beam of light will refract for incident beams not perpendicular to the laser beam entrance face 12. Exemplary materials defining the medium of the laser-transmitting machining tool 10 may include but are not limited to any type of single or poly crystal transmissive media including but not limited to: diamonds; sapphires; moissanites; chrysoberyls; alexandrite; and the like. In other configurations, exemplary materials defining the medium of the laser-transmitting machining tool 10 may include but are not limited to other transmissive media such as, for example: carbides; cubic boron nitride (CBN); silicon; nitrides; steels; alloys; ceramics; alumina; glass; glass composites; composites; and the like. The amount that light will refract is based on Snell's law, which states that the sines of the entry angles are constrained using the following relation:

$\begin{matrix} {\frac{\sin \; \theta_{1}}{\sin \; \theta_{2}} = {\frac{n_{2}}{n_{1}} = \frac{\sin \; \theta_{i}}{\sin \; \theta_{r}}}} & (4) \end{matrix}$

Assuming n₁=1 for air, θ₂ can be derived as follows:

$\begin{matrix} {{\sin \; \theta_{2}} = \frac{\sin \; \theta_{1}}{n_{2}}} & (5) \\ {\theta_{2} = {\sin^{- 1}\left( \frac{\sin \; \theta_{1}}{n_{2}} \right)}} & (6) \\ {{{For}\mspace{14mu} {small}\mspace{14mu} {angles}},{{\sin (x)} \cong x},{\therefore{\theta_{2} \cong \frac{\theta_{1}}{n_{2}}}},{{{or}\mspace{14mu} \theta_{r}} \cong \frac{\theta_{i}}{n_{2}}}} & (7) \end{matrix}$

For the triangle ABC identified at angles A, B, and C in FIG. 27, where angle A is 90°-θ_(i) and angle C is θ_(i)-θ_(r) using the alternate interior angle relationship. Using the rewritten form for Snell's law, angle C may also be rewritten as:

$\begin{matrix} {\theta_{i} - \frac{\theta_{i}}{n_{2}}} & (8) \end{matrix}$

For a desired location of the laser beam L below the line a of the cutting edge 22, the triangle ABC can be solved for the minimum back angle required to refract the laser beam upward into the cutting edge 22 using the following formula provided that the index of refraction n of the laser-transmitting machining tool 10 and length l of the laser-transmitting machining tool 10 is known (noting that that length l_(c) is the compensated length of the triangle for a reduction in length due to the back-relief angle θ_(i)). In an example, a diamond-based laser-transmitting machining tools 10 may be defined by an initial lap amount h, ranging between 0.050 mm to 0.100 mm. Therefore the corresponding inverse tangent for the length l shortened is small for

$\begin{matrix} {\theta_{i} < {20{^\circ}\mspace{14mu} {and}\mspace{14mu} {when}\mspace{14mu} \frac{l}{h_{i}}} \geq 20} & (9) \end{matrix}$

and it can be assumed that

l _(c) ≅l  (10)

Combining the approximation of equation 10 with the small-angle approximation of equation 8, equation 11, which is shown below, can be solved for known values a and l in order to obtain θ_(i).

$\begin{matrix} {a = {{l\frac{{\tan \left( {\theta_{i} - \frac{\theta_{i}}{n_{2}}} \right)} \cdot {\tan \left( {{90{^\circ}} - \theta_{i}} \right)}}{{\tan \left( {\theta_{i} - \frac{\theta_{i}}{n_{2}}} \right)} + {\tan \left( {{90{^\circ}} - \theta_{i}} \right)}}\mspace{14mu} {for}\mspace{14mu} 0} < \theta_{i} < {20{^\circ}}}} & (11) \end{matrix}$

Where.

-   -   l_(c)≅l=length of the diamond     -   a=desired location of beam below cutting edge line     -   θ_(i)=minimum angle of incidence to acheive refraction of beam         to cutting edge

With reference to FIGS. 31A and 31B, the desired location of the laser beam may correspond to the light (i.e., laser) beam diameter Φ. In an example, the desired location of the beam may directly correspond to the laser beam diameter Φ according to Equation 12, which is shown below

$\begin{matrix} {a = {\frac{\Phi}{2}\left( {1 + {R\mspace{14mu} \%}} \right)}} & (12) \end{matrix}$

where R % corresponds to the extra margin of safety to ensure the entire laser beam L is below the line of the cutting edge 22.

Utilizing Equation 11 and Equation 12 above, the following Examples and associated Tables represent a plurality of exemplary laser-transmitting machining tools 10. As seen below, each of the exemplary laser-transmitting machining tools 10 may be defined by, for example, different rake angles θ₁₄ and materials (e.g., single or poly crystal transmissive media such as diamonds, sapphires, moissanites, chrysoberyls, alexandrite, and the like, or, alternatively, other transmissive media such as carbides, cubic boron nitride (CBN), silicon, nitrides, steels, alloys, ceramics, alumina, glass, glass composites, composites, and the like) defining the medium of the laser-transmitting machining tool 10.

The following exemplary laser-transmitting machining tool 10 is directed to a negative rake angle θ₁₄ (see, e.g., FIG. 30A, 30B or 30C) and a diamond material.

Example 1

TABLE 2 R % 20% l  2.4 mm n₂ 2.417 Φ 0.200 mm h_(i) 0.050 mm

Applying the variable data of Table 2 to Equation 12, a (i.e., the desired location of the light beam below the cutting edge 22) is solved as follows:

$\begin{matrix} {a = {\frac{\Phi}{2}\left( {1 + {R\mspace{14mu} \%}} \right)}} & (13) \\ {a = {\frac{0.200}{2}\left( {1 + 0.20} \right)}} & (14) \\ {a = {0.12\mspace{14mu} {mm}}} & (15) \end{matrix}$

Whereby the effective beam position below the first side face 18 of the laser-transmitting machining tool 10 is: (h_(i)+a)=(0.050 mm+0.12 mm)=0.17 mm.

Then, applying solved a (i.e., the desired location of the light beam below the cutting edge 22) and the variable data of Table 2 to Equation 11, the minimum relief angle, θ_(i), is solved, as follows.

$\begin{matrix} {a = {{l\frac{{\tan \left( {\theta_{i} - \frac{\theta_{i}}{n_{2}}} \right)} \cdot {\tan \left( {{90{^\circ}} - \theta_{i}} \right)}}{{\tan \left( {\theta_{i} - \frac{\theta_{i}}{n_{2}}} \right)} + {\tan \left( {{90{^\circ}} - \theta_{i}} \right)}}\mspace{14mu} {for}\mspace{14mu} 0} < \theta_{i} < {20{^\circ}}}} & (16) \\ {0.12 = {{2.4\frac{{\tan \left( {\theta_{i} - \frac{\theta_{i}}{2.417}} \right)} \cdot {\tan \left( {{90{^\circ}} - \theta_{i}} \right)}}{{\tan \left( {\theta_{i} - \frac{\theta_{i}}{2.417}} \right)} + {\tan \left( {{90{^\circ}} - \theta_{i}} \right)}}\mspace{14mu} {for}\mspace{14mu} 0} < \theta_{i} < {20{^\circ}}}} & (17) \\ {\theta_{i} = {5{^\circ}}} & (18) \end{matrix}$

The following exemplary laser-transmitting machining tool 10 is directed to a negative rake angle θ₁₄ (see. e.g., FIG. 30A, 30B, or 30C) and a sapphire material.

Example 2

TABLE 3 R % 20% l  2.4 mm n₂ 1.7 Φ 0.200 mm h_(i) 0.050 mm

Applying the variable data of Table 3 to Equation 12, a (i.e., the desired location of the light beam below the cutting edge 22) is solved as follows:

$\begin{matrix} {a = {\frac{\Phi}{2}\left( {1 + {R\mspace{14mu} \%}} \right)}} & (19) \\ {a = {\frac{0.200}{2}\left( {1 + 0.20} \right)}} & (20) \\ {a = {0.12\mspace{14mu} {mm}}} & (21) \end{matrix}$

Whereby the effective beam position below the first side face 18 of the laser-transmitting machining tool 10 is: (h_(i)+a)=(0.050 mm+0.12 mm)=0.17 mm.

Then, applying solved a (i.e., the desired location of the light beam below the cutting edge 22) and the variable data of Table 3 to Equation 11, the minimum relief angle, θ_(i), is solved, as follows:

$\begin{matrix} {a = {{l\frac{{\tan \left( {\theta_{i} - \frac{\theta_{i}}{n_{2}}} \right)} \cdot {\tan \left( {{90{^\circ}} - \theta_{i}} \right)}}{{\tan \left( {\theta_{i} - \frac{\theta_{i}}{n_{2}}} \right)} + {\tan \left( {{90{^\circ}} - \theta_{i}} \right)}}\mspace{14mu} {for}\mspace{14mu} 0} < \theta_{i} < {20{^\circ}}}} & (22) \\ {0.12 = {{2.4\frac{{\tan \left( {\theta_{i} - \frac{\theta_{i}}{1.7}} \right)} \cdot {\tan \left( {{90{^\circ}} - \theta_{i}} \right)}}{{\tan \left( {\theta_{i} - \frac{\theta_{i}}{1.7}} \right)} + {\tan \left( {{90{^\circ}} - \theta_{i}} \right)}}\mspace{14mu} {for}\mspace{14mu} 0} < \theta_{i} < {20{^\circ}}}} & (23) \\ {\theta_{i} = {7{^\circ}}} & (24) \end{matrix}$

Comparatively, as seen above, the lower index of refraction n₂ defined by sapphire of EXAMPLE 2 results in a greater back-relief angle θ_(i) to direct the laser beam L to the cutting edge 22, given the same entry position of the laser beam L below the first side face 18 of the diamond-based laser-transmitting machining tool 10 of EXAMPLE 1.

The following exemplary laser-transmitting machining tool 10 is directed to a zero rake angle θ₁₄ (see, e.g., FIG. 30D) and a diamond material.

TABLE 4 R % 70% l  2.4 mm n2 2.417 Φ 0.200 mm hi    0 mm

Applying the variable data of Table 4 to Equation 12, a (i.e., the desired location of the light beam below the cutting edge 22) is solved as follows:

$\begin{matrix} {a = {\frac{\Phi}{2}\left( {1 + {R\mspace{14mu} \%}} \right)}} & (25) \\ {a = {\frac{0.200}{2}\left( {1 + 0.70} \right)}} & (26) \\ {a = {0.17\mspace{14mu} {mm}}} & (27) \end{matrix}$

Whereby the effective beam position below the first side face 18 of the laser-transmitting machining tool 10 is: (h_(i)+a)=(0 mm+0.17 mm)=0.17 mm.

Then, applying solved a (i.e, the desired location of the light beam below the cutting edge 22) and the variable data of Table 4 to Equation 11, the minimum relief angle, θ_(i), is solved, as follows:

$\begin{matrix} {a = {{l\frac{{\tan \left( {\theta_{i} - \frac{\theta_{i}}{n_{2}}} \right)} \cdot {\tan \left( {{90{^\circ}} - \theta_{i}} \right)}}{{\tan \left( {\theta_{i} - \frac{\theta_{i}}{n_{2}}} \right)} + {\tan \left( {{90{^\circ}} - \theta_{i}} \right)}}\mspace{14mu} {for}\mspace{14mu} 0} < \theta_{i} < {20{^\circ}}}} & (28) \\ {0.17 = {{2.4\frac{{\tan \left( {\theta_{i} - \frac{\theta_{i}}{2.417}} \right)} \cdot {\tan \left( {{90{^\circ}} - \theta_{i}} \right)}}{{\tan \left( {\theta_{i} - \frac{\theta_{i}}{2.417}} \right)} + {\tan \left( {{90{^\circ}} - \theta_{i}} \right)}}\mspace{14mu} {for}\mspace{14mu} 0} < \theta_{i} < {20{^\circ}}}} & (29) \\ {\theta_{i} = {7{^\circ}}} & (30) \end{matrix}$

Referring now to FIGS. 1A-26′, a plurality of exemplary optomechanical laser-transmitting machining tools are shown generally at 10 a-10 o. Each laser-transmitting machining tool 10 a-10 o includes a plurality of surfaces (at least, e.g.: a laser beam entrance face 12; a rake face 14, a flank face or clearance face 16; a first side face or a rake side face 18; and a second side face or a flank side face 20). Although each laser-transmitting machining tool 10 a-10 o includes the plurality of surfaces 12-20, some of the laser-transmitting machining tool 10 a-10 o may be further defined to include one or more additional surfaces (see, e.g., secondary clearance face 34 at FIGS. 9, 11, 13, and 15). Alternatively, in some instance, some of the laser-transmitting machining tool 10 a-10 o may not include one or more of the plurality of surfaces 12-20 (see, e.g., FIG. 15 where the laser-transmitting machining tool 10 h does not include a rake side face 18). Accordingly, each laser-transmitting machining tool 10 a-10 o is configured to provide high efficiency of laser guidance through the medium defining each laser-transmitting machining tool 10 a-10 o with minimal losses and internal reflections such that each laser-transmitting machining tool 10 a-10 o. (1) increases in determinism of cutting a workpiece W as a result of smaller losses; and (2) increases in ductility of the surface of the workpiece W being machined as a consequence of localized heating of the laser emitted from the laser-transmitting machining tool 10 a-10 o.

In some examples, the medium defining each laser-transmitting machining tool 10 a-10 o may include a single or poly crystal transmissive media such as diamonds, sapphires, moissanites, chrysoberyls, alexandrite, and the like. In other examples, the medium defining each laser-transmitting machining tool 10 a-10 o may include other transmissive media such as carbides, cubic boron nitride (CBN), silicon, nitrides, steels, alloys, ceramics, alumina, glass, glass composites, composites, and the like.

Each medium provides a different transmission rate for different wavelengths in addition to a chemical inertness to the material of the workpiece W. Therefore, the optical, chemical, and mechanical properties of a selected medium of a laser-transmitting machining tool 10 a-10 o may be exploited so that the laser-transmitting machining tool 10 a-10 o is: (1) optically transparent to the wavelength of the light source used as an assistance to a machining process; (2) mechanically harder than the material of the workpiece W; and (3) chemically inert toward the material of the workpiece W (and, if also applicable, a cutting fluid applied to the laser-transmitting machining tool 10 a-10 o).

Referring now to FIGS. 1A-2I, exemplary laser-transmitting machining tools are shown generally at 10 a. The medium of the laser-transmitting machining tools 10 a may include any desirable material such as, for example any type of single or poly crystal transmissive media including but not limited to: diamonds; sapphires; moissanites, chrysoberyls; alexandrite; and the like. In other configurations, exemplary materials defining the medium of the laser-transmitting machining tools 10 a may include but are not limited to other transmissive media such as, for example: carbides; cubic boron nitride (CBN); silicon; nitrides; steels; alloys; ceramics; alumina; glass; glass composites; composites, and the like.

The exemplary laser-transmitting machining tools 10 a are defined by a substantially similar structural configuration with respect to the transmitting machining tool 10 of FIG. 27 described above and includes a plurality of surfaces or faces 12-20. The surface 12 of the plurality of surfaces or faces 12-20 may be referred to as a laser-beam entrance face. The surface 14 of the plurality of surfaces or faces 12-20 may be referred to as a rake face. The surface 16 of the plurality of surfaces or faces 12-20 may be referred to as a flank face or clearance face. The surface 18 of the plurality of surfaces or faces 12-20 may be referred to as a first side face or a rake side face. The surface 20 of the plurality of surfaces or faces 12-20 may be referred to as a second side face or a flank side face. In some instances, the laser-transmitting machining tool 10 a may be utilized for machining a workpiece W (see, e.g., any of FIGS. 29, and 30A-30E).

Referring to FIGS. 1A-2I, the exemplary laser-transmitting machining tools 10 a are also defined a plurality of sidewall surfaces or faces 21 a-21 d. The plurality of sidewall surfaces or faces 21 a-21 d includes a first upstream sidewall surface or face 21 a (see, e.g., FIGS. 1A-G and 2A-21), a first downstream sidewall surface or face 21 b (see, e.g., FIGS. 1A-G and 2A-2I), a second upstream sidewall surface or face 21 c (see, e.g., FIGS. 2A-2I), and a second downstream sidewall surface or face 21 d (see, e.g., FIGS. 2A-21).

Each of the first upstream surface or face 21 a and the second upstream sidewall surface or face 21 c extends from the laser-beam entrance face 12. Each of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d extends from the rake face 14 and flank face or clearance face 16.

The first upstream surface or face 21 a meets the first downstream sidewall surface or face 21 b at a first side edge 23 a (see, e.g., FIGS. 1A-1G and 2A-21) that is arranged at an angle θ₂₃ (see, e.g., FIGS. 1A-G) that is substantially similar to the flank angle or clearance angle θ₁₆ that will be described in greater detail below. The first upstream surface or face 21 a and the first downstream sidewall surface or face 21 b connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 a to the second side 30 (i.e., one or both of the flank face 16 and the second side face 20) of the laser-transmitting machining tools 10 a.

The second upstream sidewall surface or face 21 c meets the second downstream sidewall surface or face 21 d at a second side edge 23 b (see, e.g., FIGS. 2A-21) that is similarly arranged at the angle θ₂₃. The second upstream surface or face 21 c and the second downstream sidewall surface or face 21 d also connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 a to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tools 10 a.

A first end 18 ₁ of the rake side face 18 extends away from a first end 12 ₁ of the laser-beam entrance face 12. A first end 20 ₁ of the flank side face 20 extends away from a second end 12 ₂ of the laser-beam entrance face 12. A first end 14 ₁ of the rake face 14 extends away from a second end 18 ₂ of the rake side face 18. A first end 16 ₁ of the flank face 16 extends away from a second end 20 of the flank side face 20. A second end 14 ₂ of the rake face 14 is joined to a second end 16 ₂ of the flank face 16 to define a cutting edge 22 that may be non-linear, curved or arcuate (as seen in, e.g., FIGS. 2A-2I); although the cutting edge 22 may be non-linear, curved or arcuate, the cutting edge 22 may be defined to include other configurations, such as, for example, a linear, non-curved or non-arcuate shape. Furthermore, the first end 14 ₁ of the rake face 14 extends away from the second end 18 ₂ of the rake side face 18 at a negative or obtuse rake angle θ₁₄; accordingly, the rake face 14 may be referred to as a negative rake face. The first end 16 ₁ of the flank face 16 extends away from the second end 20 ₂ of the flank side face 20 at an obtuse flank angle or clearance angle θ₁₆.

In some examples, the second end 12 ₂ of the laser-beam entrance face 12 extends away from the first end 20 ₁ of the second side face 20 at a back-relief angle θ₁₂. As seen at FIGS. 1A-1G, the back-relief angle θ₁₂ is obtuse (i.e., greater than 90°). In some implementations, the obtuse back-relief angle θ₁₂ of the laser-transmitting machining tool 10 a is approximately equal to 102°. However, in other examples as seen at, e.g., FIGS. 3A-3G, exemplary laser-transmitting machining tools 10 b include a back-relief angle θ₁₂ that is acute (i.e, less than 90°). In yet other examples, as seen at, e.g., FIGS. 5A-5G, exemplary laser-transmitting machining tools 10 c include a back-relief angle θ₁₂, that may be a right angle (i.e., equal to 90°).

As seen at FIGS. 1A-1G, the laser beam L that enters and then exits the laser-transmitting machining tools 10 a is shown being defined by a plurality of segments L_(E1), L_(E1), L_(E2). The plurality of segments L_(E1), L₁, L_(E2) include a laser beam entrance segment L_(E1), a laser beam refracted segment L₁ and a laser beam exit segment L_(E2). The laser beam entrance segment L_(E1) is collimated, which is generally defined by a tube or cylindrical arrays of rays (see, e.g., the laser beam L of FIGS. 31A-31B including a central ray Φ_(A) and circumferential arrays of rays Φ_(R1), Φ_(R2)).

With reference to FIGS. 1A-1G, the laser beam entrance face 12 is configured to receive and refract the collimated laser beam entrance segment L_(E1) such that the laser beam refracted segment L₁ is directed toward and through one or more of: (1) the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W). The back-relief angle θ₁₂ is configured to refract the laser beam refracted segment L₁ at the laser beam entrance face 12 according to Snell's law. The obtuse back-relief angle θ₁₂ results in the laser beam refracted segment L₁ being refracted in a direction away from the flank side face 20.

Although the laser beam entrance segment L_(E1) entering the laser beam entrance face 12 is collimated, the laser beam entrance segment L_(E1) may enter the laser beam entrance face 12 in other configurations. In some instances, the laser beam entrance segment L_(E1) may be defined by a converging laser beam (as seen in, e.g., FIGS. 5D _(C) and 5E_(C)). In other examples, the laser beam entrance segment L_(E1) may be defined by a diverging laser beam (as seen in, e.g., FIGS. 5D _(D) and 5E_(D)).

In some instances, the negative rake angle θ₁₄ and the clearance angle θ₁₆ are configured so that both the rake face 14 and the flank face 16 receive the laser beam L at an angle less than a critical angle θ_(C), as defined in Equation 3. For instance, the negative rake angle θ₁₄ may be an obtuse angle greater than 90 and less than 80°; in some instances, the rake angle θ₁₄ may range between approximately 95^(°) 115° and the clearance angle θ₁₆ may be an obtuse angle less than 105°. The back-relief angle θ₁₂ may be configured to cause the laser beam refracted segment L₁ to refract at entrance face 12 toward, respectively, the negative rake face 14 at an angle less than the critical angle or toward and the flank face 16 at an angle less than the critical angle, when the following relationship is satisfied for obtuse rake face angles θ₁₄ and flank face angles θ₁₆:

(θ₁₆−90°)+(θ₁₄−90°)<20_(C)  (31)

Referring to FIGS. 1A-1G. 1B′-1G′, and 2A-2I, a plurality of alternative laser beam entrance faces 12 defining the laser beam entrance end 24 of the laser-transmitting machining tool 10 a are shown. Referring initially to FIGS. 1A and 2A, the laser beam entrance face 12 may be substantially linear, flat or planar as the laser beam entrance face 12 extends between the first end 12 ₁ of the laser-beam entrance face 12 and so the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 a at the substantially linear, flat or planar laser beam entrance face 12, the laser beam refracted segment L₁ of the laser beam L remains collimated as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 a. After the laser beam refracted segment L₁ contacts and travels through one or more of: (1) the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), (2) the negative rake face 14 near the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent.

As seen at FIGS. 1B′ and 2B, the laser beam entrance face 12 may be defined by an inwardly-projecting (e.g., concave) surface having a non-linear, curved or arcuate configuration such as, e.g., an inwardly-projecting axial cylindrical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 a at the laser beam entrance face 12 defined by the inwardly-projecting axial cylindrical configuration, the laser beam refracted segment L₁ of the laser beam L becomes divergent (see, e.g., FIG. 2B) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 a. After the laser beam refracted segment L₁ contacts and travels through one or more of: (1) the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), (2) the negative rake face 14 near the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIGS. 1B and 2B).

Referring to FIGS. 1C′ and 2C, the laser beam entrance face 12 may be defined by an outwardly-projecting (e.g., convex) surface having a non-linear, curved or arcuate configuration such as, e.g., an axial cylindrical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 a at the laser beam entrance face 12 defined by the outwardly-projecting axial cylindrical configuration, the laser beam refracted segment L₁ of the laser beam L becomes convergent (see, e.g., FIG. 2C) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 a and then diverges from a focal point F_(P) (see, e.g., FIG. 2C) upstream up of the cutting edge 22. After the laser beam refracted segment L₁ contacts and travels through one or more of: one or more of: (1) the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), (2) the negative rake face 14 near the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIG. 1C).

As seen at FIGS. 1D′ and 2D, the laser beam entrance face 12 may be defined by an inwardly-projecting (e.g., concave) surface having a non-linear, curved or arcuate configuration such as, e.g., a lateral cylindrical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 a at the laser beam entrance face 12 defined by the inwardly-projecting lateral cylindrical configuration, the laser beam refracted segment L₁ of the laser beam L becomes divergent (see, e.g., FIG. 1D) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 a. After the laser beam refracted segment L₁ contacts and travels through one or more of: (1) the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIGS. 1D and 2D).

Referring to FIGS. 1E′ and 2E, the laser beam entrance face 12 may be defined by an outwardly-projecting (e.g., convex) surface having a non-linear, curved or arcuate configuration such as, e.g., a lateral cylindrical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 a at the laser beam entrance face 12 defined by the outwardly-projecting lateral cylindrical configuration, the laser beam refracted segment L₁ of the laser beam L becomes convergent (see, e.g., FIG. 1E) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 a and then diverges from a focal point F_(P) (see, e.g., FIG. 1E) upstream up of the cutting edge 22. After the laser beam refracted segment L contacts and travels through one or more of: (1) the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), and (3) the flank face 16 near the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIGS. 1E and 2E).

As seen at FIGS. 1F′ and 2F, the laser beam entrance face 12 may be defined by an inwardly-projecting (e.g., concave) surface having a non-linear, curved or arcuate configuration such as, e.g., a spherical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E2) enters the laser-transmitting machining tool 10 a at the laser beam entrance face 12 defined by the inwardly-projecting spherical configuration, the laser beam refracted segment L₁ of the laser beam L becomes divergent (see, e.g., FIGS. 1F and 2F) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 a. After the laser beam refracted segment L₁ contacts and travels through one or more of (1) the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIGS. 1F and 2F).

Referring to FIGS. 1G′ and 2G, the laser beam entrance face 12 may be defined by an outwardly-projecting (e.g., convex) surface having a non-linear, curved or arcuate configuration such as, e.g., a spherical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 a at the laser beam entrance face 12 defined by the outwardly-projecting spherical configuration, the laser beam refracted segment L₁ of the laser beam L becomes convergent as the laser beam refracted segment L travels through the laser-transmitting machining tool 10 a and then diverges from a focal point F_(P) (see, e.g., FIGS. 1G and 2G) upstream up of the cutting edge 22. After the laser beam refracted segment L₁ contacts and travels through one or more of: (1) the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIG. 1G).

Referring to FIG. 2H, the laser beam entrance face 12 may be defined by a combination of (1) a substantially linear, flat or planar as the laser beam entrance face 12 extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12; and (2) one or more diffractive surface portions 12D extending between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. Adjacent diffractive surface portion 12 _(D) may be spaced apart by a distance, d. Although the laser beam entrance face 12 including the one or more diffractive surface portions 12 _(D) is shown being associated with a substantially linear, flat or planar as the laser beam entrance face 12 that is substantially similar to FIGS. 1A and 2A, any of the laser beam entrance faces 12 of FIGS. 1B-1G and 2B-2G may also include the one or more diffractive surface portions 12 _(D).

At least one of the one or more diffractive surface portions 12 _(D) of the laser beam entrance face 12 receives and diffracts the laser beam entrance segment L_(E1) of the laser beam L, splitting the laser beam entrance segment L_(E1) into the plurality of diffracted laser beam segments L₁ within the body of the laser-transmitting machining tool 10 a. The plurality of diffracted laser beam segments L₁ may include three diffracted laser beam segments L₁. Each diffracted laser beam segment L₁ is separated by the diffraction angle θ_(D). The three diffracted laser beam segments L₁ distribute laser power over a total angle of 20D. Assuming n₁=1 for air, the total swept angle 2θ_(D) of the diffracted laser beams L having a wavelength λ and passing through a diffractive surface portion 12 _(D) comprising a grating with distance d between slits may be given using the grating equation to compute θ_(D): n2λ=d sin θ_(D)

n ₂ λ=d sin(θ_(D))  (32)

In some examples, at least one of the one or more diffractive surface portions 12 _(D) of the laser beam entrance face 12 is configured to diffract the laser beam entrance segment L_(E1) from a diffraction point F_(P) at a diffractive surface portion 12 _(D) of the one or more diffractive surface portions 12 _(D) such that the plurality of diffracted laser beam segments L₁ are directed toward and received by the arcuate or curved cutting edge 22.

In some examples, the one of the one or more diffractive surface portions 12 _(D) focuses or defocuses the laser beam L. or increases or decreases the focal length of the laser-transmitting machining tool 10 a. Furthermore, in some configurations, two or more laser beams L may distribute laser power more broadly on the workpiece W than a single laser beam L.

Referring to FIG. 2I, the laser beam entrance face 12 may be substantially linear, flat or planar as the laser beam entrance face 12 extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. Additionally, as seen at FIG. 2I, one or more surfaces of the plurality of surfaces 12-20 of the laser-transmitting machining tool 10 a is partially or wholly coated with a reflection-enhancing coating 36. Although the reflection-enhancing coating 36 is shown being associated with a substantially linear, flat or planar as the laser beam entrance face 12 that is substantially similar to FIGS. 1A and 2A, any of the laser-transmitting machining tools 10 a of FIGS. 1B-G and 2B-2G may also include the reflection-enhancing coating 36.

The coating 36 enhances reflection of the laser beam L (which may be defined by a converging laser beam entrance segment L_(E1) as seen at FIG. 2I) by providing a mirror surface on the laser-transmitting machining tool 10 a that reflects the laser beam L to desired zones of the laser-transmitting machining tool 10 a with the goal of an increase in efficiency and tool coverage. In some instances, the coating 36 is applied to the laser-transmitting machining tool 10 a when the ability of employing internal reflections is not feasible. In some implementations, the coating 36 includes a metallic material. Exemplary metallic materials that may be utilized for the coating 36 includes but is not limited to aluminum, silver, gold, Inconel, chrome, nickel, and titanium nitride. The coating 36 may be disposed over (1) a portion of the rake face 14 near the first end 22 ₁ of the arcuate or curved cutting edge 22; (2) a portion of the flank face 16 near the second end 22 ₂ of the arcuate or curved cutting edge 22; and (3) one or both of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d near the arcuate or curved cutting edge 22.

As seen at FIG. 2I, the laser-beam entrance face 12 is configured to receive a laser beam L. The laser beam L may be further defined by a plurality of segments L_(E1), L₁, L_(E2). The plurality of segments L_(E1), L₁, L_(E2) includes at least, for example, a converging laser beam entrance segment L_(E1), a laser beam refracted segment L₁ and a laser beam exit segment L_(E2) The laser beam entrance face 12 receives the converging laser beam entrance segment L_(E1). The laser beam refracted segment L₁ converges at a first focal point F_(P1) upstream of an outward-most portion of the arcuate or curved cutting edge 22. Thereafter the laser beam refracted segment L₁ diverges from the first focal point F_(P1) and may be incident upon one or both of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d that includes the coating 36. Thereafter, the laser beam refracted segment L₁ of the laser beam L converges at a second focal point F_(P2) upstream of the outward-most portion of the arcuate or curved cutting edge 22. Thereafter the laser beam refracted segment L₁ diverges from the second focal point F_(P2) and may be incident upon the arcuate or curved cutting edge 22. Thereafter, the laser beam refracted segment L₁ of the laser beam L is refracted off the arcuate or curved cutting edge 22 to define the laser beam exit segment L_(E2) that will then subsequently converge at a third focal point FP, downstream of the outward-most portion of the arcuate or curved cutting edge 22.

Referring now to FIGS. 3A-3G and 4A-4I, exemplary laser-transmitting machining tools are shown generally at 10 b. The medium of the laser-transmitting machining tool 10 b may include any desirable material such as, for example any type of single or poly crystal transmissive media including but not limited to: diamonds; sapphires, moissanites; chrysoberyls; alexandrite; and the like. In other configurations, exemplary materials defining the medium of the laser-transmitting machining tool 10 b may include but are not limited to other transmissive media such as, for example: carbides; cubic boron nitride (CBN); silicon; nitrides; steels; alloys; ceramics; alumina; glass; glass composites; composites; and the like.

The laser-transmitting machining tools 10 b are substantially similar to the laser-transmitting machining tools 10 a described above. Accordingly, the description associated with reference numerals of the laser-transmitting machining tools 10 b shown at FIGS. 3A-3G and 4A-4I are similar to the laser-transmitting machining tools 10 a at FIGS. 1A-G and 2A-2I; therefore, attention is directed to the written description associated with FIGS. 1A-1G and 2A-2I in order to identify structure associated with reference numerals of the laser-transmitting machining tools 10 b of FIGS. 3A-3G and 4A-41.

Unlike the back-relief angle θ₁₂ of the laser-transmitting machining tools 10 a of FIGS. 1A-1G that are arranged at an obtuse (i.e., greater than 90°) angle, the back-relief angle θ₁₂ of the laser-transmitting machining tools 10 b of FIGS. 3A-3G are arranged at an acute (i.e., less than 90°) angle. In some implementations, the acute back-relief angle θ₁₂ of the laser-transmitting machining tools 10 b may be approximately equal to 40°.

As seen at FIGS. 3A-3G, the laser beam L that enters and then exits the laser-transmitting machining tools 10 b is shown being defined by a plurality of segments L_(E1), L₁, L₂, L_(E2). The plurality of segments L_(E1), L₁, L₂, L_(E2) include a laser beam entrance segment L_(E1), a laser beam refracted segment L₁, a laser beam reflected segment L₂, and a laser beam exit segment L_(E2). The laser beam entrance segment L_(E1) is collimated, which is generally defined by a tube or cylindrical arrays of rays (see, e.g., the laser beam L of FIGS. 31A-31B including a cental ray Φ_(A) and circumferential arrays of rays Φ_(R1), Φ_(R2)).

With reference to FIGS. 3A-3G, the laser beam entrance face 12 is configured so to receive and refract the collimated laser beam entrance segment L_(E1) such that the laser beam refracted segment L₁ (that is subsequently reflected by the flank side face 20 for defining the laser beam reflected segment L₂) is directed toward and through one or more of: (1) the arcuate or curved cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W), (2) the negative rake face 14 near the arcuate or curved cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); and (3) the flank face 16 near the arcuate or curved cutting edge 22 (causing, e.g., the laser beam exit segment. La to refract onto the workpiece W). The back-relief angle θ₁₂ is configured to refract the laser beam refracted segment L₁ (that is subsequently reflected by the flank side face 20 for defining the laser beam reflected segment L₂) at the laser beam entrance face 12 according to Snell's law. Although the cutting edge 22 may be non-linear, curved or arcuate, the cutting edge 22 may be defined to include other configurations, such as, for example, a linear, non-curved or non-arcuate shape. The acute back-relief angle θ₁₂ results in the laser beam refracted segment L₁ being refracted in a direction toward the flank side face 20 for subsequently defining the laser beam reflected segment U that is reflected off of the flank side face 20.

The flank side face 20 receives the laser beam refracted segment L₁ at an incident mirror angle θ_(m-i) (that is referenced from a reference line θ_(m) extending perpendicularly away from the flank side face 20). The flank side face 20 reflects the laser beam refracted segment L to define the laser beam reflected segment L₂. The laser beam reflected segment L₂ extends away from the flank side face 20 at a reflected mirror angle θ_(m-r) (that is also referenced from the reference line θ_(m) extending perpendicularly away from the flank side face 20). The reflected mirror angle θ_(m-r) is equal to the incident mirror angle θ_(m-i).

Although the laser beam entrance segment L_(E1) entering the laser beam entrance face 12 is collimated, the laser beam entrance segment L_(E1) may enter the laser beam entrance face 12 in other configurations. In some instances, the laser beam entrance segment L_(E1) may be defined by a converging laser beam (as seen in, e.g., FIGS. 5D _(C) and 5E_(C)). In other examples, the laser beam entrance segment L_(E1) may be defined by a diverging laser beam (as seen in, e.g., FIGS. 5D _(D) and 5E_(D)).

In some instances, the negative rake angle θ₁₄ and the clearance angle θ₁₆ are configured so that both the rake face 14 and the flank face 16 receive the laser beam L at an angle less than a critical angle θ_(C), as defined in Equation 3. For instance, the negative rake angle θ₁₄ may be an obtuse angle greater than 90° and less than 180°; in some instances, the rake angle θ₁₄ may range between approximately 95° and 115° and the clearance angle θ₁₆ may be an obtuse angle less than 105°. The back-relief angle θ₁₂ may be configured to cause the laser beam refracted segment L₁ to refract at entrance face 12 toward, respectively, the negative rake face 14 at an angle less than the critical angle or toward and the flank face 16 at an angle less than the critical angle, when the following relationship is satisfied for obtuse rake face angles θ₁₄ and flank face angles θ₁₆:

(θ₁₆−90°)+(θ₁₄−90°)<2θ_(C)  (33)

Referring to FIGS. 3A-3G. 3B′-3G′, and 4A-4I, a plurality of alternative laser beam entrance faces 12 defining the laser beam entrance end 24 of the laser-transmitting machining tool 10 b are shown. Referring initially to FIGS. 3A and 4A, the laser beam entrance face 12 may be substantially linear, flat or planar as the laser beam entrance face 12 extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 b at the substantially linear, flat or planar laser beam entrance face 12, the laser beam refracted segment L₁ of the laser beam L remains collimated as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 b. After the laser beam reflected segment L₂ is subsequently reflected by the flank side face 20 and contacts and travels through one or more of (1) the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); (2) the negative rake face 14 near the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); and (3) the flank face 16 near the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent.

As seen at FIGS. 3B′ and 4B, the laser beam entrance face 12 may be defined by an inwardly-projecting (e.g., concave) surface having a non-linear, curved or arcuate so configuration such as, e.g., an inwardly-projecting axial cylindrical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 b at the laser beam entrance face 12 defined by the inwardly-projecting axial cylindrical configuration, the laser beam refracted segment L₁ of the laser beam L becomes divergent (see, e.g., FIG. 4B) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 b. After the laser beam reflected segment L₂ is subsequently reflected by the flank side face 20 and contacts and travels through one or more of: (1) the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); (2) the negative rake face 14 near the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W), and (3) the flank face 16 near the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIGS. 3B and 4B).

Referring to FIGS. 3C′ and 4C, the laser beam entrance face 12 may be defined by an outwardly-projecting (e.g., convex) surface having a non-linear, curved or arcuate configuration such as, e.g., an axial cylindrical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 b at the laser beam entrance face 12 defined by the outwardly-projecting axial cylindrical configuration, the laser beam refracted segment L₁ of the laser beam L becomes convergent (see, e.g., FIG. 4C) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 b and then diverges from a focal point FP (see, e.g., FIG. 4C) upstream up of the arcuate or curved cutting edge 22. After the laser beam reflected segment L₂ is subsequently reflected by the flank side face 20 and contacts and travels through one or more of: one or more of: (1) the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); (2) the negative rake face 14 near the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); and (3) the flank face 16 near the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIG. 3C.

As seen at FIGS. 3D′ and 4D, the laser beam entrance face 12 may be defined by an inwardly-projecting (e.g., concave) surface having a non-linear, curved or arcuate configuration such as, e.g., a lateral cylindrical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 b at the laser beam entrance face 12 defined by the inwardly-projecting lateral cylindrical configuration, the laser beam refracted segment L of the laser beam L becomes divergent (see. e.g., FIG. 3D) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 b. After the laser beam reflected segment L₂ is subsequently reflected by the flank side face 20 and contacts and travels through one or more of: (1) the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); (2) the negative rake face 14 near the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); and (3) the flank face 16 near the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIGS. 3D and 4D).

Referring to FIGS. 3E′ and 4E, the laser beam entrance face 12 may be defined by an outwardly-projecting (e.g., convex) surface having a non-linear, curved or arcuate configuration such as, e.g., a lateral cylindrical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 b at the laser beam entrance face 12 defined by the outwardly-projecting lateral cylindrical configuration, the laser beam refracted segment L₁ of the laser beam L becomes convergent (see, e.g., FIG. 3E) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 b and then diverges from a focal point F_(P) (see, e.g., FIG. 3E) upstream up of the arcuate or curved cutting edge 22. After the laser beam reflected segment L₂ is subsequently reflected by the flank side face 20 and contacts and travels through one or more of: (1) the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); (2) the negative rake face 14 near the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); and (3) the flank face 16 near the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIGS. 3E and 4E).

As seen at FIGS. 3F′ and 4F, the laser beam entrance face 12 may be defined by an inwardly-projecting (e.g., concave) surface having a non-linear, curved or arcuate configuration such as, e.g., a spherical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L₁ enters the laser-transmitting machining tool 10 b at the laser beam entrance face 12 defined by the inwardly-projecting spherical configuration, the laser beam refracted segment L₁ of the laser beam L becomes divergent (see, e.g., FIGS. 3F and 4F) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 b. After the laser beam reflected segment L₂ is subsequently reflected by the flank side face 20 and contacts and travels through one or more of: (1) the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); (2) the negative rake face 14 near the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); and (3) the flank face 16 near the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIGS. 3F and 4F).

Referring to FIGS. 3G′ and 4G, the laser beam entrance face 12 may be defined by an outwardly-projecting (e.g., convex) surface having a non-linear, curved or arcuate configuration such as, e.g., a spherical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 b at the laser beam entrance face 12 defined by the outwardly-projecting spherical configuration, the laser beam refracted segment L₁ of the laser beam L becomes convergent as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 b and then diverges from a focal point F_(P) (see. e.g., FIGS. 3G and 4G) upstream up of the arcuate or curved cutting edge 22. After the laser beam reflected segment L₂ is subsequently reflected by the flank side face 20 and contacts and travels through one or more of: (1) the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); (2) the negative rake face 14 near the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); and (3) the flank face 16 near the arcuate or curved cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L remains divergent (see, e.g., FIGS. 3G and 4G).

Referring to FIG. 4H, the laser beam entrance face 12 may be defined by a combination of: (1) a substantially linear, flat or planar as the laser beam entrance face 12 extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12; and (2) one or more diffractive surface portions 12 _(D) extending between the first end 12 of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. Adjacent diffractive surface portion 12 _(D) may be spaced apart by a distance, d. Although the laser beam entrance face 12 including the one or more diffractive surface portions 12D is shown being associated with a substantially linear, flat or planar as the laser beam entrance face 12 that is substantially similar to FIGS. 3A and 4A, any of the laser beam entrance faces 12 of FIGS. 3B-3G and 4B-4G may also include the one or more diffractive surface portions 12D.

At least one of the one or more diffractive surface portions 12D of the laser beam entrance face 12 receives and diffracts the laser beam entrance segment LEI of the laser beam L, splitting the laser beam entrance segment L_(E1) into the plurality of diffracted laser beam segments L₁ within the body of the laser-transmitting machining tool 10 b. The plurality of diffracted laser beam segments L₁ may include three diffracted laser beam segments L₁. Each diffracted laser beam segment L₁ is separated by the diffraction angle θ_(D). The three diffracted laser beam segments L₁ distribute laser power over a total angle of 2θ_(D). Assuming n₁=1 for air, the total swept angle 2θ_(D) of the diffracted laser beams L having a wavelength λ and passing through a diffractive surface portion 12 _(D) so comprising a grating with distance d between slits may be given using the grating equation to compute θ_(D): n2λ=d sin θ_(D)

n ₂ λ=d sin(θ_(D))  (34)

In some examples, at least one of the one or more diffractive surface portions 12 _(D) of the laser beam entrance face 12 is configured to diffract the laser beam entrance segment L_(E1) from a diffraction point F_(P) at a diffractive surface portion 12 _(D) of the one or more diffractive surface portions 12 _(D) such that the plurality of diffracted laser beam segments L₁ are directed toward and received (as laser beam reflected segments L₂ reflected by the flank side face 20) by the arcuate or curved cutting edge 22.

In some examples, the one of the one or more diffractive surface portions 12 _(D) focuses or defocuses the laser beam L, or increases or decreases the focal length of the laser-transmitting machining tool 10 b. Furthermore, in some configurations, two or more laser beams L may distribute laser power more broadly in the workpiece W than a single laser beam L.

Referring to FIG. 4I, the laser beam entrance face 12 may be substantially linear, flat or planar as the laser beam entrance face 12 extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. Additionally, as seen at FIG. 4I, one or more surfaces of the plurality of surfaces 12-20 of the laser-transmitting machining tool 10 b is partially or wholly coated with a reflection-enhancing coating 36. Although the reflection-enhancing coating 36 is shown being associated with a substantially linear, flat or planar as the laser beam entrance face 12 that is substantially similar to FIGS. 3A and 4A, any of the laser-transmitting machining tools 10 b of FIGS. 3B-3G and 4B-4G may also include the reflection-enhancing coating 36.

The coating 36 enhances reflection of the laser beam L (which may be defined by a converging laser beam entrance segment L_(E1) as seen at FIG. 4I) by providing a mirror surface on the laser-transmitting machining tool 10 b that reflects the laser beam L to desired zones of the laser-transmitting machining tool 10 b with the goal of an increase in efficiency and tool coverage. In some instances, the coating 36 is applied to the laser-transmitting machining tool 10 b when the ability of employing internal reflections is not feasible. In some implementations, the coating 36 includes a metallic material. Exemplary metallic materials that may be utilized for the coating 36 includes but is not limited to aluminum, silver, gold, Inconel, chrome, nickel, and titanium nitride. The coating 36 may be disposed over (1) a portion of the rake face 14 near the first end 22 ₁ of the arcuate or curved cutting edge 22; (2) a portion of the flank face 16 near the second end 22 ₂ of the arcuate or curved cutting edge 22, and (3) one or both of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d near the arcuate or curved cutting edge 22.

As seen at FIG. 4I, the laser-beam entrance face 12 is configured to receive a laser beam L. The laser beam L may be further defined by a plurality of segments L_(E1). L₁, L_(E2). The plurality of segments L_(E1), L₁, L₂, and L_(E2) includes at least, for example, a converging laser beam entrance segment L_(E1), a laser beam refracted segment L₁, a laser beam reflected segment L₂, a laser beam exit segment L_(E2). The laser beam entrance face 12 receives the converging laser beam entrance segment L_(E1). The laser beam refracted segment L₁ converges at a first focal point F_(P1) upstream of an outward-most portion of the arcuate or curved cutting edge 22. Thereafter the laser beam refracted segment L₁ diverges from the first focal point F_(P1) and may (as laser beam reflected segments L₂ reflected by the flank side face 20) be incident upon one or both of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d that includes the coating 36. Thereafter, the laser beam reflected segments L₂ of the laser beam L converges at a second focal point F_(P2) upstream of the outward-most portion of the arcuate or curved cutting edge 22. Thereafter the laser beam reflected segments L₂ diverges from the second focal point F_(P2) and may be incident upon the arcuate or curved cutting edge 22. Thereafter, the laser beam reflected segments L₂ of the laser beam L is refracted off the arcuate or curved cutting edge 22 to define the laser beam exit segment L_(E2) that will then subsequently converge at a third focal point F_(P3) downstream of the outward-most portion of the arcuate or curved cutting edge 22.

Referring now to FIGS. 5A-5G and 6A-6 exemplary laser-transmitting machining tools are shown generally at 10 c The medium of the laser-transmitting machining tools 10 c may include any desirable material such as, for example any type of so single or poly crystal transmissive media including but not limited to: diamonds; sapphires; moissanites; chrysoberyls; alexandrite; and the like. In other configurations, exemplary materials defining the medium of the laser-transmitting machining tools 10 c may include but are not limited to other transmissive media such as, for example: carbides; cubic boron nitride (CBN); silicon; nitrides; steels; alloys; ceramics; alumina, glass; glass composites; composites; and the like.

The laser-transmitting machining tools 10 c are substantially similar to the laser-transmitting machining tools 10 a described above. Accordingly, the description associated with reference numerals of the laser-transmitting machining tools 10 c shown at FIGS. 5A-5G and 6A-6I are similar to the laser-transmitting machining tools 10 a at FIGS. 1A-1G and 2A-2I, therefore, attention is directed to the written description associated with FIGS. 1A-1G and 2A-2I in order to identify structure associated with reference numerals of the laser-transmitting machining tools 10 c of FIGS. 5A-5G and 6A-6I.

Unlike the back-relief angle θ₁₂ of the laser-transmitting machining tools 10 a of FIGS. 1A-G and 2A-2I that are arranged at an obtuse (i.e., greater than 90°) angle, the back-relief angle θ₁₂ of the laser-transmitting machining tools 10 c of FIGS. 5A-5G are arranged at a right (i.e., equal to 90°) angle.

As seen at FIGS. 5A-5G, the laser beam L that enters and then exits the laser-transmitting machining tools 10 c is shown being defined by a plurality of segments U, L₁, L_(E2). The plurality of segments L_(E1), L₁, L_(E2) include a laser beam entrance segment L_(E1), a laser beam segment L₁ and a laser beam exit segment L_(E2). The laser beam entrance segment L_(E1) is collimated, which is generally defined by a tube or cylindrical arrays of rays (see, e.g., the laser beam L of FIGS. 31A-31B including a central ray Φ_(A) and circumferential arrays of rays Φ_(R1), Φ_(R2)).

With reference to FIGS. 5A-5G, the laser beam entrance face 12 is configured to receive the collimated laser beam entrance segment L_(E1) such that the laser beam segment L₁ is directed toward and through one or more of: (1) the arcuate or curved cutting edge 22 (causing, e.g., the laser beam exit segment L₁ to refract onto the workpiece W); (2) the negative rake face 14 near the arcuate or curved cutting edge 22 (causing. e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); and (3) the flank face 16 near the arcuate or curved cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W). Although the cutting edge 22 may be non-linear, curved or arcuate, the cutting edge 22 may be defined to include other configurations, such as, for example, a linear, non-curved or non-arcuate shape. The back-relief angle θ₁₂ is configured to receive the laser beam segment L₁ at the laser beam entrance face 12 according to Snell's law. As seen at FIGS. 5A and 6A, the perpendicular or right back-relief angle θ₁₂ of an exemplary substantially linear, flat or planar as the laser beam entrance face 12 results in the laser-transmitting machining tool 10 c not refracting the laser beam entrance segment L_(E1); rather, the laser beam entrance face 12 of the laser-transmitting machining tool 10 c of FIGS. 5A and 6A permits the laser beam entrance segment L_(E1) to pass into the body of the transmitting machining tool 10 c for defining the laser beam segment L₁.

Although the laser beam entrance segment L_(E1) entering the laser beam entrance face 12 is collimated, the laser beam entrance segment L_(E1) may enter the laser beam entrance face 12 in other configurations. In some instances, the laser beam entrance segment LEI may be defined by a converging laser beam (as seen in, e.g., FIGS. 5D _(C) and 5E_(C)). In other examples, the laser beam entrance segment L_(E1) may be defined by a diverging laser beam (as seen in, e.g., FIGS. 5D _(D) and 5E_(D)).

In some instances, the negative rake angle θ₁₄ and the clearance angle θ₁₆ are configured so that both the rake face 14 and the flank face 16 receive the laser beam L at an angle less than a critical angle θ_(C), as defined in Equation 3. For instance, the negative rake angle θ₁₄ may be an obtuse angle greater than 90° and less than 180°; in some instances, the rake angle θ₁₄ may range between approximately 95° 115° and the clearance angle θ₁₄ may be an obtuse angle less than 105°. The back-relief angle θ₁₂ may be configured to receive the laser beam segment L₁ at entrance face 12 for subsequently being directed toward, respectively, the negative rake face 14 at an angle less than the critical angle or toward and the flank face 16 at an angle less than the critical angle, when the following relationship is satisfied for obtuse rake face angles θ₁₄ and flank face angles θ₁₆:

(θ₁₆−90°)+(θ₁₄−90°)<2θ_(C)  (35)

Referring to FIGS. 5A-5G. 5B′-5G′, and 6A-6I, a plurality of alternative laser beam entrance faces 12 defining the laser beam entrance end 24 of the laser-transmitting machining tools 10 c are shown. Referring initially to FIGS. 5A and 6A, the laser beam entrance face 12 may be substantially linear, flat or planar as the laser beam entrance face 12 extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 c at the substantially linear, flat or planar laser beam entrance face 12, the laser beam segment L₁ of the laser beam L remains collimated as the laser beam segment L₁ travels through the laser-transmitting machining tool 10 c. After the laser beam segment L₁ contacts and travels through one or more of: (1) the arcuate or curved cutting edge 22 (causing the laser beam segment L₁ to refract onto the workpiece W), (2) the negative rake face 14 near the arcuate or curved cutting edge 22 (causing the laser beam segment L₁ to refract onto the workpiece W); and (3) the flank face 16 near the arcuate or curved cutting edge 22 (causing the laser beam segment L₁ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIG. 6A).

As seen at FIGS. 5B′ and 6B, the laser beam entrance face 12 may be defined by an inwardly-projecting (e.g., concave) surface having a non-linear, curved or arcuate configuration such as, e.g., an inwardly-projecting axial cylindrical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 c at the laser beam entrance face 12 defined by the inwardly-projecting axial cylindrical configuration, the laser beam refracted segment L₁ of the laser beam L becomes divergent (see, e.g., FIG. 6B) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 c. After the laser beam refracted segment L₁ contacts and travels through one or more of: (1) the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); (2) the negative rake face 14 near the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); and (3) the flank face 16 near the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIG. 6B).

Referring to FIGS. 5C′ and 6C, the laser beam entrance face 12 may be defined by an outwardly-projecting (e.g., convex) surface having a non-linear, curved or arcuate configuration such as, e.g., an axial cylindrical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 c at the laser beam entrance face 12 defined by the outwardly-projecting axial cylindrical configuration, the laser beam refracted segment L₁ of the laser beam L becomes convergent (see, e.g., FIG. 6C) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 c and then diverges from a focal point F (see, e.g., FIG. 6C) upstream up of the arcuate or curved cutting edge 22. After the laser beam refracted segment L₁ contacts and travels through one or more of: one or more of: (1) the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); (2) the negative rake face 14 near the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); and (3) the flank face 16 near the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L remains divergent (see, e.g., FIG. 6C).

As seen at FIGS. 5D′ and 6D, the laser beam entrance face 12 may be defined by an inwardly-projecting (e.g., concave) surface having a non-linear, curved or arcuate configuration such as, e.g., a lateral cylindrical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L₁ enters the laser-transmitting machining tool 10 c at the laser beam entrance face 12 defined by the inwardly-projecting lateral cylindrical configuration, the laser beam refracted segment L₁ of the laser beam L becomes divergent (see, e.g., FIG. 5D) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 c. After the laser beam refracted segment L₁ contacts and travels through one or more of: (1) the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); (2) the negative rake face 14 near the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); and (3) the flank face 16 near the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIG. 6D).

With reference to FIG. 5D _(C), unlike the example described above associated with FIGS. 5D, 5D′, and 6D, the laser beam entrance segment L_(E1) may be defined by a converging laser beam. The inwardly-projecting lateral cylindrical configuration of the laser beam entrance face 12 receives the laser beam entrance segment L_(E1) that is defined by a converging cone angle θ_(CA) having a first focal point F_(P); accordingly, the laser beam L may be referred to as a convergent laser beam. The first focal point F_(P) is located downstream from of the inwardly-projecting lateral cylindrical configuration of the laser beam entrance face 12.

The inwardly-projecting lateral cylindrical configuration of the laser beam entrance face 12 refracts the laser beam L according to Snell's law, causing the laser beam refracted segment L₁ to have a transformed cone angle θ_(CA)′ and a transformed focal point F_(P)′. The inwardly-projecting lateral cylindrical configuration of the laser beam entrance face 12 may focus the laser beam L such that the transformed focal point F_(P)′ is arranged within the laser-transmitting machining tool 10 c, upstream of the rake face 14 and the flank face 16 at a second distance D2 extending from the focal point F_(P).

With reference to the following equation, the second distance D2 associated with the transformed focal point F_(P)′ may be dictated by the following equation, where positive values of D1 extend upstream, and positive values of D2 extend downstream:

$\begin{matrix} {{D\; 2} = \frac{n^{2}{RD}\; 1}{{\left( {n_{2} - n_{1}} \right)D\; 1} - {n_{1}R}}} & (36) \end{matrix}$

Accordingly, the transformed cone angle θ_(CA)′ is related to the cone angle θ_(CA) by the following equation:

tan(θ_(CA)′/2)=abs(D1/D2)tan(θ_(CA)/2)  (37)

In another example as seen at FIG. 5D _(D), unlike the example described above associated with FIG. FIG. 5D _(C), the laser beam entrance segment L_(E1) may be defined by a diverging laser beam. The inwardly-projecting lateral cylindrical configuration of the laser beam entrance face 12 receives the laser beam entrance segment L_(E1) that is defined by a diverging cone angle θ_(CA) having a first focal point F_(P); accordingly, the laser beam L may be referred to as a divergent laser beam. The first focal point F_(P) is located upstream of the inwardly-projecting lateral cylindrical configuration of the laser beam entrance face 12. The inwardly-projecting lateral cylindrical configuration of the laser beam entrance face 12 refracts the laser beam L according to Snell's law, causing the laser beam refracted segment L₁ to have a transformed cone angle θ_(CA)′; because the laser beam entrance segment L_(E1) is defined by a diverging laser beam, the laser beam entrance segment L_(E1) does not define a transformed focal point F_(P)′, and, as such, the laser beam entrance segment L_(E1) continues to diverge as it travels through the laser-transmitting machining tool 10 c toward the rake face 14, the flank face 16, and the arcuate or curved cutting edge 22.

Referring to FIGS. 5E′ and 6E, the laser beam entrance face 12 may be defined by an outwardly-projecting (e.g., convex) surface having a non-linear, curved or arcuate configuration such as, e.g., a lateral cylindrical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 c at the laser beam entrance face 12 defined by the outwardly-projecting lateral cylindrical configuration, the laser beam refracted segment L₁ of the laser beam L becomes convergent (see, e.g., FIG. 5E) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 c and then diverges from a focal point FP (see, e.g., FIG. 5E) upstream up of the arcuate or curved cutting edge 22. After the laser beam refracted segment L₁ contacts and travels through one or more of: (1) the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L to refract onto the workpiece W); (2) the negative rake face 14 near so the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); and (3) the flank face 16 near the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), the laser beam exit segment L of the laser beam L becomes convergent (see, e.g., FIG. 6E).

With reference to FIG. 5E _(C), unlike the example described above associated with FIGS. 5E, 5E′, and 6E, the laser beam entrance segment LE may be defined by a converging laser beam. The outwardly-projecting lateral cylindrical configuration of the laser beam entrance face 12 receives the laser beam entrance segment L_(E1) that is defined by a converging cone angle θ_(CA) having a first focal point F_(P); accordingly, the laser beam L may be referred to as a convergent laser beam. The first focal point FP is located downstream from of the outwardly-projecting lateral cylindrical configuration of the laser beam entrance face 12.

The outwardly-projecting lateral cylindrical configuration of the laser beam entrance face 12 refracts the laser beam L according to Snell's law, causing the laser beam refracted segment L₁ to have a transformed cone angle θ_(CA)′ and a transformed focal point F_(P)′. The outwardly-projecting lateral cylindrical configuration of the laser beam entrance face 12 may focus the laser beam L such that the transformed focal point F_(P)′ is arranged within the laser-transmitting machining tool 10 c, upstream of the rake face 14 and the flank face 16 at a second distance D2 extending from the focal point F_(P).

With reference to the following equation, the second distance D2 associated with the transformed focal point F_(P)′ may be dictated by the following equation, where positive values of D1 extend upstream, and positive values of D2 extend downstream.

$\begin{matrix} {{D\; 2} = \frac{n^{2}{RD}\; 1}{{\left( {n_{2} - n_{1}} \right)D\; 1} - {n_{1}R}}} & (38) \end{matrix}$

Accordingly, the transformed cone angle θ_(CA)′ is related to the cone angle θ_(CA) by the following equation:

tan(θ_(CA)′/2)=abs(D1/D2)tan(θ_(CA)/2)  (39)

In another example as seen at FIG. 5E _(D), unlike the example described above associated with FIG. FIG. 5E _(C), the laser beam entrance segment L_(E1) may be defined by a diverging laser beam. The outwardly-projecting lateral cylindrical configuration of the laser beam entrance face 12 receives the laser beam entrance segment L_(E1) that is defined by a diverging cone angle θ_(CA) having a first focal point F_(P); accordingly, the laser beam L may be referred to as a divergent laser beam. The first focal point F_(P) is located upstream of the outwardly-projecting lateral cylindrical configuration of the laser beam entrance face 12. The outwardly-projecting lateral cylindrical configuration of the laser beam entrance face 12 refracts the laser beam L according to Snell's law, causing the laser beam refracted segment L₁ to have a transformed cone angle θ_(CA)′, because the laser beam entrance segment L_(E1) is defined by a diverging laser beam, the outwardly-projecting lateral cylindrical configuration of the laser beam entrance face 12 may defocus the laser beam L such that the transformed focal point F_(P)′ is arranged outside of the laser-transmitting machining tool 10 c, downstream of the rake face 14, the flank face 16, and the arcuate or curved cutting edge 22 at a second distance D2 extending from the outwardly-most portion of the outwardly-projecting lateral cylindrical configuration of the laser beam entrance face 12, and, as such, the laser beam entrance segment L_(E1) converges as it travels through the laser-transmitting machining tool 10 c toward the rake face 14, the flank face 16, and the arcuate or curved cutting edge 22.

As seen at FIGS. 5F′ and 6F, the laser beam entrance face 12 may be defined by an inwardly-projecting (e.g., concave) surface having a non-linear, curved or arcuate configuration such as, e.g., a spherical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 c at the laser beam entrance face 12 defined by the inwardly-projecting spherical configuration, the laser beam refracted segment L₁ of the laser beam L becomes divergent (see, e.g., FIGS. 5F and 6F) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 c. After the laser beam refracted segment L₁ contacts and travels through one or more of: (1) the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), (2) the negative rake face 14 near the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L_(1t) to refract onto the workpiece W); and (3) the flank face 16 near the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIG. 6F).

Referring to FIGS. 5G′ and 6G, the laser beam entrance face 12 may be defined by an outwardly-projecting (e.g., convex) surface having a non-linear, curved or arcuate configuration such as, e.g., a spherical configuration that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 c at the laser beam entrance face 12 defined by the outwardly-projecting spherical configuration, the laser beam refracted segment L₁ of the laser beam L becomes convergent as the laser beam refracted segment L travels through the laser-transmitting machining tool 10 c and then diverges from a focal point F_(P) (see, e.g., FIGS. 5G and 6G) upstream up of the arcuate or curved cutting edge 22. After the laser beam refracted segment L₁ contacts and travels through one or more of: (1) the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); (2) the negative rake face 14 near the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W); and (3) the flank face 16 near the arcuate or curved cutting edge 22 (causing the laser beam refracted segment L₁ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L remains divergent (see, e.g., FIGS. 5G and 6G).

Referring to FIG. 6H, the laser beam entrance face 12 may be defined by a combination of: (1) a substantially linear, flat or planar as the laser beam entrance face 12 extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12, and (2) one or more diffractive surface portions 12 _(D) extending between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. Adjacent diffractive surface portions 12 _(D) may be spaced apart by a distance, d. Although the laser beam entrance face 12 including the one or more diffractive surface portions 12 _(D) is shown being associated with a substantially linear, flat or planar as the laser beam entrance face 12 that is substantially similar to FIGS. 5A and 6A, any of the laser beam entrance faces 12 of FIGS. 58-5G and 6B-6G may also include the one or more diffractive surface portions 12 _(D).

At least one of the one or more diffractive surface portions 12 _(D) of the laser beam entrance face 12 receives and diffracts the laser beam entrance segment L_(E1) of the laser beam L, splitting the laser beam entrance segment L into the plurality of diffracted laser beam segments L₁ within the body of the laser-transmitting machining tool 10 c. The plurality of diffracted laser beam segments L₁ may include three diffracted laser beam segments L₁. Each diffracted laser beam segment L₁ is separated by the diffraction angle θ_(D). The three diffracted laser beam segments L₁ distribute laser power over a total angle of 2θ_(D). Assuming n₁=1 for air, the total swept angle 2θ_(D) of the diffracted laser beams L having a wavelength λ and passing through a diffractive surface portion 12D comprising a grating with distance d between slits may be given using the grating equation to compute θ_(D): n2λ=d sin θ_(D)

n ₂ λ=d sin(θ_(D))  (40)

In some examples, at least one of the one or more diffractive surface portions 12 _(D) of the laser beam entrance face 12 is configured to diffract the laser beam entrance segment LE from a diffraction point FP at a diffractive surface portion 12 _(D) of the one or more diffractive surface portions 12 _(D) such that the plurality of diffracted laser beam segments L₁ are directed toward and received by the arcuate or curved cutting edge 22.

In some examples, the one of the one or more diffractive surface portions 12 _(D) focuses or defocuses the laser beam L, or increases or decreases the focal length of the laser-transmitting machining tool 10 c. Furthermore, in some configurations, two or more laser beams L may distribute laser power more broadly in the workpiece W than a single laser beam L.

Referring to FIG. 6I, the laser beam entrance face 12 may be substantially linear, flat or planar as the laser beam entrance face 12 extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. Additionally, as seen at FIG. 6I, one or more surfaces of the plurality of surfaces 12-20 of the laser-transmitting machining tool 10 c is partially or wholly coated with a reflection-enhancing coating 36. Although the reflection-enhancing coating 36 is shown being associated with a substantially linear, flat or planar as the laser beam entrance face 12 that is substantially similar to FIGS. 5A and 6A, any of the laser-transmitting machining tools 10 c of FIGS. 5B-5G and 6B-6G may also include the reflection-enhancing coating 36.

The coating 36 enhances reflection of the laser beam L (which may be defined by a converging laser beam entrance segment L_(E1) as seen at FIG. 6I) by providing a mirror surface on the laser-transmitting machining tool 10 c that reflects the laser beam L to desired zones of the laser-transmitting machining tool 10 c with the goal of an increase in efficiency and tool coverage. In some instances, the coating 36 is applied to the laser-transmitting machining tool 10 c when the ability of employing internal reflections is not feasible. In some implementations, the coating 36 includes a metallic material. Exemplary metallic materials that may be utilized for the coating 36 includes but is not limited to aluminum, silver, gold, Inconel, chrome, nickel, and titanium nitride. The coating 36 may be disposed over: (1) a portion of the rake face 14 near the first end 22 of the arcuate or curved cutting edge 22; (2) a portion of the flank face 16 near the second end 22 ₂ of the arcuate or curved cutting edge 22; and (3) one or both of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d near the arcuate or curved cutting edge 22.

As seen at FIG. 6I, the laser-beam entrance face 12 is configured to receive a laser beam L. The laser beam L may be further defined by a plurality of segments L_(E1), L₁, L_(E2). The plurality of segments L_(E1), L₁, L_(E2) includes at least, for example, a converging laser beam entrance segment L_(E1), a laser beam refracted segment L₁ and a laser beam exit segment L_(E2). The laser beam entrance face 12 receives the converging laser beam entrance segment L_(E1). The laser beam refracted segment L₁ converges at a first focal point F_(P1) upstream of an outward-most portion of the arcuate or curved cutting edge 22. Thereafter the laser beam refracted segment L₁ diverges from the first focal point F_(P1) and may be incident upon one or both of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d that includes the coating 36. Thereafter, the laser beam refracted segment L₁ of the laser beam L converges at a second focal point F_(P2) upstream of the outward-most portion of the arcuate or curved cutting edge 22. Thereafter the laser beam refracted segment L diverges from the second focal point F_(P2) and may be incident upon the arcuate or curved cutting edge 22. Thereafter, the laser beam refracted segment L₁ of the laser beam L is refracted off the arcuate or curved cutting edge 22 to define the laser beam exit segment L that will then subsequently converge at a third focal point FP, downstream of the outward-most portion of the arcuate or curved cutting edge 22.

Referring now to FIGS. 7A-7G and 8A-8I, exemplary laser-transmitting machining tools are shown generally at 10 d. The medium of the laser-transmitting machining tools 10 d may include any desirable material such as, for example any type of single or poly crystal transmissive media including but not limited to: diamonds; sapphires; moissanites; chrysoberyls; alexandrite; and the like. In other configurations, exemplary materials defining the medium of the laser-transmitting machining tools 10 d may include but are not limited to other transmissive media such as, for example: carbides; cubic boron nitride (CBN), silicon, nitrides, steels; alloys; ceramics, alumina; glass; glass composites; composites; and the like.

Referring to FIGS. 7A-8I, the exemplary laser-transmitting machining tools 10 d are also defined a plurality of sidewall surfaces or faces 21 a-21 d. The plurality of sidewall surfaces or faces 21 a-21 d includes a first upstream sidewall surface or face 21 a (see, e.g., FIGS. 7A-7G and 8A-8I), a first downstream sidewall surface or face 21 b (see, e.g., FIGS. 7A-7G and 8A-81), a second upstream sidewall surface or face 21 c (see, e.g., FIGS. 8A-8I), and a second downstream sidewall surface or face 21 d (see, e.g. FIGS. 8A-8I).

Each of the first upstream surface or face 21 a and the second upstream sidewall surface or face 21 c extends from the laser-beam entrance face 12. Each of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d extends from the rake face 14 and flank face or clearance face 16.

The first upstream surface or face 21 a meets the first downstream sidewall surface or face 21 b at a first side edge 23 a (see, e.g., FIGS. 7A-7G and 8A-8I) that is arranged at an angle θ₂₃ (see, e.g., FIGS. 7A-7G) that is substantially similar to the flank angle or clearance angle θ₁₆ that will be described in greater detail below. The first upstream surface or face 21 a and the first downstream sidewall surface or face 21 b S5 connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 d to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tools 10 d.

The second upstream sidewall surface or face 21 c meets the second downstream sidewall surface or face 21 d at a second side edge 23 b (see, e.g., FIGS. 8A-8I) that is similarly arranged at the angle θ₂₃. The second upstream surface or face 21 c and the second downstream sidewall surface or face 21 d also connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 d to the second side 30 (i.e., one or both of the flank face 16 and the second side face 20) of the laser-transmitting machining tools 10 d.

A first end 18 ₁ of the rake side face 18 extends away from a first end 12 ₁ of the laser-beam entrance face 12. A first end 20 ₁ of the flank side face 20 extends away from a second end 12 ₂ of the laser-beam entrance face 12. A first end 14 ₁ of the rake face 14 extends away from a second end 18 ₂ of the rake side face 18. A first end 16 ₁ of the flank face 16 extends away from a second end 20 ₂ of the flank side face 20. A second end 14 ₂ of the rake face 14 is joined to a second end 16 ₂ of the flank face 16 to define a cutting edge 22 that may be non-linear, curved or arcuate (as seen in, e.g., FIGS. 8A-8I); although the cutting edge 22 may be non-linear, curved or arcuate, the cutting edge 22 may be defined to include other configurations, such as, for example, a linear, non-curved or non-arcuate shape. Furthermore, the first end 14 ₁ of the rake face 14 extends away from the second end 18 ₂ of the rake side face 18 at a negative or obtuse rake angle θ₁₄; accordingly, the rake face 14 may be referred to as a negative rake face. The first end 16 of the flank face 16 extends away from the second end 20 ₂ of the flank side face 20 at an obtuse flank angle or clearance angle θ₁₆. The negative rake angle θ₁₄ may be an obtuse angle greater than 90° and less than 180°; in some instances, the rake angle θ₁₄ may range between approximately 135° and 155°.

The exemplary laser-transmitting machining tools 10 d are defined by a substantially similar structural configuration with respect to the transmitting machining tools 10 and 10 a of FIGS. 1 and 27 described above and includes the plurality of surfaces or faces 12-20 and the cutting edge 22. In some instances, the laser-transmitting machining tools 10 d may be utilized for machining a workpiece W (see, e.g., any of FIGS. 29, and 30A-30E). Furthermore, the first end 14 ₁ of the rake face 14 extends away from the second end 18 ₂ of the rake side face 18 at a negative or obtuse rake angle θ₁₄; accordingly, the rake face 14 may be referred to as a negative rake face. The first end 16 ₁ of the flank face 16 extends away from the second end 20 ₂ of the flank side face 20 at an obtuse flank angle or clearance angle θ₁₆.

The laser beam entrance face 12 may be defined by a functional entrance face segment 12 _(f) and a non-functional entrance face segment 12 _(n). A first end 12 _(n1) of the non-functional entrance face segment 12 _(n) extends perpendicularly away from the first end 20 ₁ of the flank side face 20 at a height H_(n). A first end 12 _(f1) of the functional entrance face segment 12 _(f) extends away from the second end 12 _(n2) of the non-functional entrance face segment 12 _(n) (according to a reference line associated with the height H_(n) that is parallel to the flank side face 20). A second end 12 _(f2) of the functional entrance face segment 12 _(f) extends away from the first end 18 ₁ of the rake side face 18.

The first end 12 _(f1) of the functional entrance face segment 12 _(f) extends away from the second end 12 _(n2) of the non-functional entrance face segment 12 _(n) at an angle (see, e.g., θ₁₂) that is reference from a plane (see, e.g., a dashed line) that is parallel to the flank side face 20. The angle θ₁₂ may be alternatively referred to as the back-relief angle θ₁₂. As similarly described above in, for example, FIGS. 1A-1G, in some examples, the back-relief angle θ₁₂ acute (i.e., less than 90°). In some implementations, the acute back-relief angle θ₁₂ approximately equal to 60°. The back-relief angle θ₁₂ is configured to refract the laser beam L at the functional entrance face segment 12 _(f) of the laser beam entrance face 12 according to Snell's law. The acute back-relief angle θ₁₂ results in the laser beam L being refracted in a direction away from the rake side face 18.

With reference to FIGS. 7A-7G, the functional entrance face segment 12 _(f) of the laser beam entrance face 12 is configured to receive and refract the laser beam L at a height H_(i) that is greater than the height H_(n) defining the non-functional entrance face segment 12 _(n) of the laser beam entrance face 12. Accordingly, the laser beam L does not enter the non-functional entrance face 12 _(n).

As seen at FIGS. 7A-7G, the laser beam L that enters and then exits the laser-transmitting machining tools 10 d is shown being defined by a plurality of segments L_(E1), L₁, L₂, L_(E2). The plurality of segments L_(E1), L₁, L₂, L_(E2) include a laser beam entrance segment L_(E1), a laser beam refracted segment L₁, a laser beam reflected segment L₂ and a laser beam exit segment L_(E2). The laser beam entrance segment L_(E1) is collimated, which is generally defined by a tube or cylindrical arrays of rays (see, e.g., the laser beam L of FIGS. 31A-31B including a central ray Φ_(A) and circumferential arrays of rays Φ_(R1), Φ_(R2)).

With reference to FIGS. 7A-7G, the functional entrance face segment 12 _(f) of the laser beam entrance face 12 is configured to receive and refract the collimated laser beam entrance segment LE such that the laser beam refracted segment L₁ (that is subsequently reflected by the flank side face 20 for defining the laser beam reflected segment L₂) is directed toward and through one or more of: (1) the cutting edge 22 (causing, e.g. the laser beam exit segment L_(E2) to refract onto the workpiece W), (2) the negative rake face 14 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W), and (3) the flank face 16 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W). The back-relief angle θ₁₂ is configured to refract the laser beam refracted segment L₁ at the functional entrance face segment 12 _(f) of the laser beam entrance face 12 according to Snell's law. The acute back-relief angle θ₁₂ results in the laser beam refracted segment L₁ being refracted in a direction toward the flank side face 20.

Although the laser beam entrance segment L_(E1) entering the functional entrance face segment 12 _(f) of the laser beam entrance face 12 is collimated, the laser beam entrance segment L_(E1) may enter the functional entrance face segment 12 _(f) of the laser beam entrance face 12 in other configurations. In some instances, the laser beam entrance segment L_(E1) may be defined by a converging laser beam (as seen in, e.g., FIGS. 5D _(C) and 5E_(C)). In other examples, the laser beam entrance segment L_(E1) may be defined by a diverging laser beam (as seen in, e.g., FIGS. 5D _(D) and 5E_(D)).

In some examples, the functional entrance face 12 _(f) of the laser beam entrance face 12 receives the collimated laser beam entrance segment L_(E1) at the height H_(i) above the flank side face 20; the height H_(i) is a portion of a tool height H_(t) extending between the flank side face 20 and the rake side face 18. The functional entrance face segment 12 _(f) of the laser beam entrance face 12 refracts the collimated laser beam entrance segment L_(E1) to define the laser beam refracted segment L₁ that is directed toward and received by the flank side face 20.

The flank side face 20 receives the laser beam refracted segment L₁ at an incident mirror angle θ_(m-i) (that is referenced from a reference line θ_(m) extending perpendicularly away from the flank side face 20). The flank side face 20 reflects the laser beam refracted segment L to define the laser beam reflected segment L₂. The laser beam reflected segment L₂ extends away from the flank side face 20 at a reflected mirror angle θ_(m-r) (that is also referenced from the reference line θ_(m) extending perpendicularly away from the flank side face 20). The reflected mirror angle θ_(m-r) equal to the incident mirror angle θ_(m-i).

The laser beam reflected segment L₂ is then directed toward and received by one or more of: (1) the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W). The laser beam exit segment L_(E2) may be refracted into the workpiece W with a high level of heat efficiency.

The laser beam exit segment L_(E2) exits the laser-transmitting machining tool 10 d at a height H_(e) above the flank side face 20. In some examples, the rake face receives the laser beam reflected segment L₂ at an angle less than a critical angle θ_(C), as defined in Equation 3, if the following relationship is satisfied:

abs(θ_(m)+θ₁₄−180°)<θ_(C)  (41)

As can be readily seen from FIGS. 7A-7G, the length l of the laser-transmitting machining tool 10 d may be decreased by increasing the height H_(n) defining the non-functional entrance face segment 12 _(n). The length l of the laser-transmitting machining tool 10 d may be dictated by the following equation.

l=(H _(i) +H _(e))*tan(θ_(m))+(H _(i) −H _(n))*tan(90°−θ₁₂)  (42)

Accordingly, the greater the height H. of the non-functional entrance face 12 _(n), the shorter the length l of the laser-transmitting machining tool 10 d.

Referring to FIGS. 7A-7G. 7B′-7G′, and 8A-I, a plurality of alternative laser beam entrance faces 12 defining the laser beam entrance end 24 of the laser-transmitting machining tools 10 d are shown. Referring initially to FIGS. 7A and 8A, the functional entrance face segment 12 _(f) of the laser beam entrance face 12 may be substantially linear, flat or planar as the functional entrance face segment 12 _(f) of the laser beam entrance face 12 extends between the first end 12 _(f1) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12 and the second end 12 _(f2) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12. After the collimated laser beam entrance segment LEI enters the laser-transmitting machining tool 10 d at the substantially linear, flat or planar the functional entrance face segment 12 _(f) of the laser beam entrance face 12, the laser beam refracted segment L₁ of the laser beam L remains collimated as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 d. After the laser beam reflected segment L₂ is subsequently reflected by the flank side face 20 and contacts and travels through one or more of: (1) the cutting edge 22 (causing the laser beam reflected segment L₁ to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent.

As seen at FIGS. 7B′ and 7B, the functional entrance face segment 12 _(f) of the laser beam entrance face 12 may be defined by an inwardly-projecting (e.g., concave) surface having a non-linear, curved or arcuate configuration such as, e.g., an inwardly-projecting axial cylindrical configuration that extends between the first end 12 _(f1) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12 and the second end 12 _(f2) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12. After the collimated laser beam entrance segment LEI enters the laser-transmitting machining tool 10 d at the functional entrance face segment 12 _(f) of the laser beam entrance face 12 defined by the inwardly-projecting axial cylindrical configuration, the laser beam refracted segment L₁ of the laser beam L becomes divergent (see, e.g., FIG. 8B) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 d. After the laser beam reflected segment L₂ is subsequently reflected by the flank side face 20 and contacts and travels through one or more of: (1) the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIGS. 7B and 8B).

Referring to FIGS. 7C′ and 8C, the functional entrance face segment 12 _(f) of the laser beam entrance face 12 may be defined by an outwardly-projecting (e.g., convex) surface having a non-linear, curved or arcuate configuration such as, e.g., an axial cylindrical configuration that extends between the first end 12 _(f1) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12 and the second end 12 _(f2) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 d at the functional entrance face segment 12 _(f) of the laser beam entrance face 12 defined by the outwardly-projecting axial cylindrical configuration, the laser beam refracted segment L₁ of the laser beam L becomes convergent (see, e.g., FIG. 8C) as the laser beam refracted segment L travels through the laser-transmitting machining tool 10 d and then diverges from a focal point F_(P) (see, e.g., FIG. 8C) upstream up of the cutting edge 22. After the laser beam reflected segment L₂ is subsequently reflected by the flank side face 20 and contacts and travels through one or more of: one or more of: (1) the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W), and (3) the flank face 16 near the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIG. 7C).

As seen at FIGS. 7D′ and 8D, the functional entrance face segment 12 _(f) of the laser beam entrance face 12 may be defined by an inwardly-projecting (e.g., concave) surface having a non-linear, curved or arcuate configuration such as, e.g., a lateral cylindrical configuration that extends between the first end 12 _(f1) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12 and the second end 12 _(f2) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12. Ater the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 d at the functional entrance face segment 12 _(f) of the laser beam entrance face 12 defined by the inwardly-projecting lateral cylindrical configuration, the laser beam refracted segment L₁ of the laser beam L becomes divergent (see, e.g., FIG. 7D) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 d. After the laser beam reflected segment L₂ is subsequently reflected by the flank side face 20 and contacts and travels through one or more of: (1) the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see, e.g., FIGS. 7D and 8D).

Referring to FIGS. 7E′ and 8E, the functional entrance face segment 12 _(f) of the laser beam entrance face 12 may be defined by an outwardly-projecting (e.g., convex) surface having a non-linear, curved or arcuate configuration such as, e.g., a lateral cylindrical configuration that extends between the first end 12 _(f1) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12 and the second end 12 _(f2) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12. Ater the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 d at the functional entrance face segment 12 _(f) of the laser beam entrance face 12 defined by the outwardly-projecting lateral cylindrical configuration, the laser beam refracted segment L₁ of the laser beam L becomes convergent (see, e.g., FIG. 7E) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 d and then diverges from a focal point F_(P) (see, e.g., FIG. 7E) upstream up of the cutting edge 22. Ater the laser beam reflected segment L₂ is subsequently reflected by the flank side face 20 and contacts and travels through one or more of (1) the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes convergent (see. e.g., FIGS. 7E and 8E).

As seen at FIGS. 7F′ and 8F, the functional entrance face segment 12 _(f) of the laser beam entrance face 12 may be defined by an inwardly-projecting (e.g., concave) surface having a non-linear, curved or arcuate configuration such as, e.g., a spherical configuration that extends between the first end 12 _(f1) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12 and the second end 12 _(f2) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 d at the functional entrance face segment 12 _(f) of the laser beam entrance face 12 defined by the inwardly-projecting spherical configuration, the laser beam refracted segment L₁ of the laser beam L becomes divergent (see, e.g., FIGS. 7F and 8F) as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 d. After the laser beam reflected segment L₂ is subsequently reflected by the flank side face 20 and contacts and travels through one or more of: (1) the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W), the laser beam exit segment LE of the laser beam L becomes convergent (see, e.g., FIGS. 7F and 5F).

Referring to FIGS. 7G′ and 8G, the functional entrance face segment 12 _(f) of the laser beam entrance face 12 may be defined by an outwardly-projecting (e.g., convex) surface having a non-linear, curved or arcuate configuration such as, e.g., a spherical configuration that extends between the first end 12 _(f1) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12 and the second end 12 _(f2) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12. After the collimated laser beam entrance segment L_(E1) enters the laser-transmitting machining tool 10 d at the functional entrance face segment 12 _(f) of the laser beam entrance face 12 defined by the outwardly-projecting spherical configuration, the laser beam refracted segment L₁ of the laser beam L becomes convergent as the laser beam refracted segment L₁ travels through the laser-transmitting machining tool 10 d and then diverges from a focal point F_(P) (see, e.g., FIGS. 7G and 8G) upstream up of the cutting edge 22. After the laser beam reflected segment L₂ is subsequently reflected by the flank side face 20 and contacts and travels through one or more of: (1) the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing the laser beam reflected segment L₂ to refract onto the workpiece W), the laser beam exit segment L_(E2) of the laser beam L becomes divergent (see, e.g., FIGS. 7G and 8G).

Referring to FIG. 8H, the functional entrance face segment 12 _(f) of the laser beam entrance face 12 may be defined by a combination of: (1) a substantially linear, flat or planar as the functional entrance face segment 12 _(f) of the laser beam entrance face 12 extends between the first end 12 _(f1) of the functional entrance face segment 12 _(f1) of the laser-beam entrance face 12 and the second end 12 _(f2) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12; and (2) one or more diffractive surface portions 12 _(D) extending between the first end 12 _(f1) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12 and the second end 12 _(f2) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12. Adjacent diffractive surface portion 12 _(D) may be spaced apart by a distance, d. Although the functional entrance face segment 12 _(f) of the laser beam entrance face 12 including the one or more diffractive surface portions 12 _(D) is shown being associated with a substantially linear, flat or planar as the functional entrance face segment 12 _(f) of the laser beam entrance face 12 that is substantially similar to FIGS. 7A and 8A, any of the laser beam entrance faces 12 of FIGS. 7B-7G and 8B-8G may also include the one or more diffractive surface portions 12 _(D).

At least one of the one or more diffractive surface portions 12 _(D) of the functional entrance face segment 12 _(f) of the laser beam entrance face 12 receives and diffracts the laser beam entrance segment L_(E1) of the laser beam L, splitting the laser beam entrance segment L_(E1) into the plurality of diffracted laser beam segments L₁ within the body of the laser-transmitting machining tool 10 d. The plurality of diffracted laser beam segments L₁ may include three diffracted laser beam segments L₁. Each diffracted laser beam segment L₁ is separated by the diffraction angle θ_(D). The three diffracted laser beam segments L₁ distribute laser power over a total angle of 2θ_(D). Assuming n₁=1 for air, the total swept angle 2θ_(D) of the diffracted laser beams L having a wavelength λ and passing through a diffractive surface portion 12 _(D) comprising a grating with distance d between slits may be given using the grating equation to compute θ_(D): n2λ=d sin θ_(D)

n ₂ λ=d sin(θ_(D))  (43)

In some examples, at least one of the one or more diffractive surface portions 12 _(D) of the functional entrance face segment 12 _(f) of the laser beam entrance face 12 is configured to diffract the laser beam entrance segment L_(E1) from a diffraction point F_(P) at a diffractive surface portion 12 _(D) of the one or more diffractive surface portions 12 _(D) such that the plurality of diffracted laser beam segments L₁ are directed toward and received (as laser beam reflected segments L₂ reflected by the flank side face 20) by the arcuate or curved cutting edge 22.

In some examples, the one of the one or more diffractive surface portions 12D focuses or defocuses the laser beam L, or increases or decreases the focal length of the laser-transmitting machining tool 10 d. Furthermore, in some configurations, two or more laser beams L may distribute laser power more broadly in the workpiece W than a single laser beam L.

Referring to FIG. 8I, the functional entrance face segment 12 _(f) of the laser beam entrance face 12 may be substantially linear, flat or planar as the functional entrance face segment 12 _(f) of the laser beam entrance face 12 extends between the first end 12 _(f1) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12 and the second end 12 _(f2) of the functional entrance face segment 12 _(f) of the laser-beam entrance face 12. Additionally, as seen at FIG. 8I, one or more surfaces of the plurality of surfaces 12-20 of the laser-transmitting machining tool 10 d is partially or wholly coated with a reflection-enhancing coating 36. Although the reflection-enhancing coating 36 is shown being associated with a substantially linear, flat or planar as the functional entrance face segment 12 _(f) of the laser beam entrance face 12 that is substantially similar to FIGS. 7A and 8A, any of the laser-transmitting machining tools 10 d of FIGS. 7B-7G and 8B-G may also include the reflection-enhancing coating 36.

The coating 36 enhances reflection of the laser beam L (which may be defined by a converging laser beam entrance segment L_(E1) as seen at FIG. 8I) by providing a mirror surface on the laser-transmitting machining tool 10 d that reflects the laser beam L to desired zones of the laser-transmitting machining tool 10 d with the goal of an increase in efficiency and tool coverage. In some instances, the coating 36 is applied to the laser-transmitting machining tool 10 d when the ability of employing internal reflections is not feasible. In some implementations, the coating 36 includes a metallic material. Exemplary metallic materials that may be utilized for the coating 36 includes but is not limited to aluminum, silver, gold, Inconel, chrome, nickel, and titanium nitride. The coating 36 may be disposed over. (1) a portion of the rake face 14 near the first end 22 ₁ of the arcuate or curved cutting edge 22; (2) a portion of the flank face 16 near the second end 22 ₂ of the arcuate or curved cutting edge 22; and (3) one or both of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d near the arcuate or curved cutting edge 22.

As seen at FIG. 8I, the laser-beam entrance face 12 is configured to receive a laser beam L. The laser beam L may be further defined by a plurality of segments L_(E1), L₁, L_(E2). The plurality of segments L_(E1), L₁, L_(E2) includes at least, for example, a converging laser beam entrance segment L_(E1), a laser beam refracted segment L₁, a laser beam reflected segments L₂ and a laser beam exit segment L_(E2). The functional entrance face segment 12 _(f) of laser beam entrance face 12 receives the converging laser beam entrance segment L_(E1). The laser beam refracted segment L₁ converges at a first focal point F_(P1) upstream of an outward-most portion of the arcuate or curved cutting edge 22. Thereafter the laser beam refracted segment L₁ diverges from the first focal point F_(P1) and may (as laser beam reflected segments L₂ reflected by the flank side face 20) be incident upon one or both of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d that includes the coating 36. Thereafter, the laser beam reflected segments L₂ of the laser beam L converges at a second focal point F_(P2) upstream of the outward-most portion of the arcuate or curved cutting edge 22. Thereafter the laser beam reflected segments L₂ diverges from the second focal point F_(P2) and may be incident upon the arcuate or curved cutting edge 22. Thereafter, the laser beam reflected segments L₂ of the laser beam L is refracted off the arcuate or curved cutting edge 22 to define the laser beam exit segment L_(E2) that will then subsequently converge at a third focal point F_(P3) downstream of the outward-most portion of the arcuate or curved cutting edge 22.

Referring now to FIGS. 9 and 10, an exemplary laser-transmitting machining tool is shown generally at 10 e. The medium of the laser-transmitting machining tool 10 e may include any desirable material such as, for example any type of single or poly crystal transmissive media including but not limited to: diamonds, sapphires, moissanites; chrysoberyls; alexandrite; and the like. In other configurations, exemplary materials defining the medium of the laser-transmitting machining tool 10 e may include but are not limited to other transmissive media such as, for example: carbides; cubic boron nitride (CBN); silicon; nitrides; steels; alloys; ceramics; alumina; glass; glass composites; composites; and the like.

Referring to FIGS. 9 and 10, the exemplary laser-transmitting machining tool 10 e is also defined a plurality of sidewall surfaces or faces 21 a-21 d. The plurality of sidewall surfaces or faces 21 a-21 d includes a first upstream sidewall surface or face 21 a (see, e.g., FIGS. 9 and 10), a first downstream sidewall surface or face 21 b (see, e.g., FIGS. 9 and 10), a second upstream sidewall surface or face 21 c (see, e.g., FIG. 10), and a second downstream sidewall surface or face 21 d (see, e.g., FIG. 10). Each of the first upstream surface or face 21 a and the second upstream sidewall surface or face 21 c extends from the laser-beam entrance face 12. Each of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d extends from the rake face 14 and the flank face or clearance face 16. The first upstream surface or face 21 a meets the first downstream sidewall surface or face 21 b at a first side edge 23 a (see, e.g., FIGS. 9 and 10) that is arranged at an angle θ₂₃ (see, e.g., FIG. 9) that may be substantially similar to the flank angle or clearance angle θ₁₆ that will be described in greater detail below. The first upstream surface or face 21 a and the first downstream sidewall surface or face 21 b connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 e to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tool 10 e. The second upstream sidewall surface or face 21 c meets the second downstream sidewall surface or face 21 d at a second side edge 23 b (see, e.g., FIG. 10) that is similarly arranged at the angle θ₂₃. The second upstream surface or face 21 c and the second downstream sidewall surface or face 21 d also connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 e to the second side 30 (i.e., one or both of the flank face 16 and the second side face 20) of the laser-transmitting machining tool 10 e.

The exemplary laser-transmitting machining tool 10 e is defined by a substantially similar structural configuration with respect to the transmitting machining tools 10 and 10 a-10 d of FIGS. 1A-8I and 27 described above and includes: the plurality of surfaces or faces 12-20; the first end 12 ₁-20 ₁ of each respective surface 12-20; and the second end 12 ₂-20 ₂ of each respective surface 12-20. Furthermore, the first end 14 ₁ of the rake face 14 extends away from the second end 18 ₂ of the rake side face 18 at a negative rake angle θ₁₄; accordingly, the rake face 14 may be referred to as a negative rake face.

In some examples, the second end 12 ₂ of the laser-beam entrance face 12 extends away from the first end 20 ₁ of the second side face 20 at a back-relief angle θ₁₂. As seen at FIG. 9, the back-relief angle θ₁₂ is obtuse (i.e., greater than 90°). In some implementations, the obtuse back-relief angle θ₁₂ of the laser-transmitting machining tool 10 e is approximately equal to 102°. However, in other examples as seen at, e.g., FIG. 11, an exemplary laser-transmitting machining tool 10 f includes a back-relief angle θ₁₂ that is acute (i.e., less than 90°). In yet other examples, as seen at, e.g., FIG. 13, an exemplary laser-transmitting machining tool 10 g includes a back-relief angle θ₁₂, that may be a right angle (i.e., equal to 90°).

As seen at FIG. 9, the laser beam L that enters and then exits the laser-transmitting machining tool 10 e is shown being defined by a plurality of segments L_(E1), L₁, L_(E2). The plurality of segments L_(E1), L₁, L_(E2) include a laser beam entrance segment L_(E1), a laser beam refracted segment L₁, and a laser beam exit segment L_(E2). The laser beam entrance segment L_(E1) is collimated, which is generally defined by a tube or cylindrical arrays of rays (see, e.g., the laser beam L of FIGS. 31A-31B including a central ray Φ_(A) and circumferential arrays of rays Φ_(R1), Φ_(R2)).

With reference to FIG. 9, the laser beam entrance face 12 is configured to receive and refract the collimated laser beam entrance segment L_(E1) such that the laser beam refracted segment L₁ is directed toward and through one or more of: (1) the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W). The cutting edge 22 may be non-linear, curved or arcuate (as seen in, e.g., FIG. 10); although the cutting edge 22 may be non-linear, curved or arcuate, the cutting edge 22 may be defined to include other configurations, such as, for example, a linear, non-curved or non-arcuate shape. The back-relief angle θ₁₂ is configured to refract the laser beam refracted segment L₁ at the laser beam entrance face 12 according to Snell's law. The obtuse back-relief angle θ₁₂ results in the laser beam refracted segment L₁ being refracted in a direction away from the flank side face 20.

Although the laser beam entrance segment L_(E1) entering the laser beam entrance face 12 is collimated, the laser beam entrance segment L_(E1) may enter the laser beam entrance face 12 in other configurations. In some instances, the laser beam entrance segment L_(E1) may be defined by a converging laser beam (as seen in, e.g., FIGS. 5D _(C) and 5E_(C)). In other examples, the laser beam entrance segment L_(E1) may be defined by a diverging laser beam (as seen in, e.g., FIGS. 5D _(D) and 5E_(D)).

The laser-transmitting machining tool 10 e further includes a secondary clearance face 34 extending between and connecting the flank face 16 to the flank side face 20. In some examples, a first end 34 ₁ of the secondary clearance face 34 extends away from the second end 20 ₂ of the flank side face 20, and a second end 34 ₂ of the secondary clearance face 34 extends away from the first end 16 ₁ of the flank face 16. In some instances, the laser-transmitting machining tool 10 e may be utilized for machining a workpiece W (see, e.g., any of FIGS. 29, and 30A-30E).

The first end 16 ₁ of the flank face 16 extends away from the second end 34 ₂ of the secondary clearance face 34 to define a flank angle or clearance angle θ₁₆. The flank angle or clearance angle θ₁₆ is obtuse.

The secondary clearance face 34 may extend away from the flank side face 20 so at a secondary clearance angle θ₃₄. In some instances, the secondary clearance angle θ₃₄ is obtuse. The secondary clearance face angle θ₃₄ may be between 120° and 180°. In some implementations, the obtuse secondary clearance angle θ₃₄ is approximately equal to 150°. In some instances, assuming that the secondary clearance angle θ₃₄ is approximately equal to 120°, the clearance angle θ₁₆ may range between approximately 110° and approximately 130°.

As seen at FIG. 9, the laser beam entrance face 12 receives the laser beam entrance segment L_(E1) at a height H_(i) above the flank side face 20; the height H_(i) is a portion of a tool height H_(t) extending between the flank side face 20 and the rake side face 18. The laser beam entrance face 12 refracts the laser beam entrance segment L_(E1) to define the laser beam refracted segment L₁ that is directed toward and received by the rake side face 18.

The rake side face 18 receives the laser beam refracted segment L₁ at an incident mirror angle θ_(m-i) relative to the rake side face 18 (that is referenced from a reference line θ_(m) extending perpendicularly away from the rake side face 18). The rake side face 18 reflects the laser beam refracted segment L₁ to define the laser beam reflected segment L₂. The laser beam reflected segment L₂ extends away from the rake side face 18 at a reflected mirror angle θ_(m-r) (that is also referenced from the reference line θ_(m) extending perpendicularly away from the rake side face 18). The reflected mirror angle θ_(m-r) is equal to the incident mirror angle θ_(m-i).

The laser beam reflected segment L₂ is then directed toward and received by one or more of: (1) the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); according to an exemplary configuration of the laser-transmitting machining tool 10 e seen at FIG. 9, the laser beam reflected segment L₂ is then directed toward and received by the second end 16 ₂ of flank face 16 near the cutting edge 22. Thereafter, the laser beam reflected segment L₂ exits the laser-transmitting machining tool 10 e and defines the laser beam exit segment L_(E2) of the laser beam L. The laser beam exit segment L_(E2) may be refracted into the workpiece W. In some instances, the laser-transmitting machining tool 10 e is used for machining a deeply concave workpiece W, such as a workpiece W having features with a small radius of curvature (ROC) and small clear aperture (CA).

The laser beam exit segment L_(E2) exits the laser-transmitting machining tool 10 e at a height H_(e) above the flank side face 20. In some examples, the rake side face 18 receives the laser beam reflected segment L₂ at an angle less than a critical angle θ_(C), as defined in Equation 3, when the following relationship is satisfied:

abs(θ_(m)−θ₃₄−θ₁₆₊₃₆₀°)<_(C),θ_(m)>θ_(C)  (44)

Although the rake side face 18 is shown reflecting the laser beam reflected segment L₂ toward the flank face 16 at FIG. 9, in other examples, the rake side face 18 may reflect the laser beam reflected segment L₂ toward the rake face 14, causing the rake face 14 to refract the laser beam reflected segment L₂ into the workpiece W. The rake side face 18 may reflect the laser beam reflected segment L₂ toward other faces of the laser-transmitting machining tool 10 e as well. The tool length l of the laser-transmitting machining tool 10 e may be dictated by the following equation:

l=(H _(t) −H _(e))*tan(θ_(m))+(H _(t) −H _(i))*tan(θ₁₂−90°)  (45)

Referring to FIGS. 9 and 10, the laser beam entrance face 12 may be defined by a substantially linear, flat or planar as the laser beam entrance face 12 that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. Although the laser beam entrance face 12 is defined by a substantially linear, flat or planar as the laser beam entrance face 12 that is substantially similar to the substantially linear, flat or planar as the laser beam entrance faces 12 of the laser-transmitting machining tools 10 a, 10 b, 10 c, and 10 d of, respectively. FIGS. 2A, 4A, 6A, and 8A, any of the other configurations described above (e.g., an inwardly-projecting axial cylindrical configuration, an outwardly-projecting axial cylindrical configuration, an inwardly-projecting lateral cylindrical configuration, an outwardly-projecting lateral so cylindrical configuration, an inwardly-projecting spherical configuration, an outwardly-projecting spherical configuration, or one or more diffractive surface portions) at FIGS. 2B-2H, 4B-4H, 6B-6H, and 8B-8H may define the laser beam entrance faces 12 of the laser-transmitting machining tools 10 e. Similarly, one or more surfaces of the plurality of surfaces 12-20 of the laser-transmitting machining tool 10 e may be partially or wholly coated with a reflection-enhancing coating 36 as similarly described above at FIGS. 2I, 4I, 6I, and 8I.

Referring now to FIGS. 11A-12A and 11B-12B, exemplary laser-transmitting machining tools are shown generally at 10 f. The medium of the laser-transmitting machining tools 10 f may include any desirable material such as, for example any type of single or poly crystal transmissive media including but not limited to: diamonds; sapphires; moissanites, chrysoberyls, alexandrite; and the like. In other configurations, exemplary materials defining the medium of the laser-transmitting machining tools 10 f may include but are not limited to other transmissive media such as, for example: carbides; cubic boron nitride (CBN); silicon; nitrides; steels; alloys; ceramics; alumina; glass; glass composites; composites; and the like.

The exemplary laser-transmitting machining tools 10 f are also defined a plurality of sidewall surfaces or faces 21 a-21 d. The plurality of sidewall surfaces or faces 21 a-21 d includes a first upstream sidewall surface or face 21 a (see, e.g., FIGS. 11A-12A and 11B-12B), a first downstream sidewall surface or face 21 b (see, e.g., FIGS. 11A-12A and 11B-12B) a second upstream sidewall surface or face 21 c (see, e.g., FIGS. 12A and 12B), and a second downstream sidewall surface or face 21 d (see, e.g., FIGS. 12A and 12B). Each of the first upstream surface or face 21 a and the second upstream sidewall surface or face 21 c extends from the laser-beam entrance face 12. Each of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d extends from the rake face 14 and the flank face or clearance face16 The first upstream surface or face 21 a meets the first downstream sidewall surface or face 21 b at a first side edge 23 a (see, e.g., FIGS. 11A-12A and 11B-12B) that is arranged at an angle θ₂₃ (see, e.g., FIGS. 11A and 11B) that is substantially similar to the flank angle or clearance angle θ₁₆ that will be described in greater detail below. The first upstream surface or face 21 a and the first downstream sidewall surface or face 21 b connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tools 10 f to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tools 10 f. The second upstream sidewall surface or face 21 c meets the second downstream sidewall surface or face 21 d at a second side edge 23 b (see, e.g., FIGS. 12A and 12B) that is similarly arranged at the angle θ₂₃. The second upstream surface or face 21 c and the second downstream sidewall surface or face 21 d also connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 f to the second side 30 (i.e, one or both of the flank face 16 and the second side face 20) of the laser-transmitting machining tools 10 f.

The exemplary laser-transmitting machining tools 10 f are defined by a substantially similar structural configuration with respect to the transmitting machining tools 10 and 10 a-10 d of FIGS. 1A-8I and 27 described above and includes: the plurality of surfaces or faces 12-20, the first end 12 ₁-20 ₁ of each respective surface 12-20; and the second end 12 ₂-20 ₂ of each respective surface 12-20. Furthermore, the first end 14 ₁ of the rake face 14 extends away from the second end 18 ₂ of the rake side face 18 at a negative rake angle θ₁₄; accordingly, the rake face 14 may be referred to as a negative rake face. The negative rake angle θ₁₄ may be an obtuse angle greater than 90° and less than 180°. In some instances, the rake angle θ₁₄ of FIG. 11A may range between approximately 135° and 155°. In other examples, the rake angle θ₁₄ of FIG. 11B may range between approximately 155° and an amount less than 180°.

In some examples, the second end 12 ₂ of the laser-beam entrance face 12 extends away from the first end 20 ₁ of the second side face 20 at a back-relief angle θ₁₂. As seen at FIGS. 11A and 11B, the back-relief angle θ₁₂ is acute (i.e., less than 90°). In some implementations as seen at, e.g., FIG. 11A, the acute back-relief angle θ₁₂ of the laser-transmitting machining tool 10 f is approximately equal to 60°. In other implementations as seen at, e.g., FIG. 11B, the acute back-relief angle θ₁₂ of the laser-transmitting machining tool 10 f is approximately equal to 70°.

As seen at FIGS. 11A and 11B, the laser beam L that enters and then exits the laser-transmitting machining tool 10 f is shown being defined by a plurality of segments L_(E1), L₁, L₂, L₂. The plurality of segments L_(E1), L₁, L₂, L₂ include a laser beam entrance segment L_(E1), a laser beam refracted segment L₁, laser beam reflected segment L₂, and a law beam exit segment L_(E2). The laser beam entrance segment L_(E1) is collimated, which is generally defined by a tube or cylindrical arrays of rays (see. e.g., the laser beam L of FIGS. 31A-31B including a central ray Φ_(A) and circumferential arrays of rays Φ_(R1), Φ_(R2)).

With reference to FIGS. 11A and 11B, the laser beam entrance face 12 is configured to receive and refract the collimated laser beam entrance segment L_(E1) such that the laser beam refracted segment L₁ is directed toward and through one or more of: (1) the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing, e.g., the laser beam exit segment LE to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W). The cutting edge 22 may be non-linear, curved or arcuate (as seen in, e.g., FIGS. 12A and 12B); although the cutting edge 22 may be non-linear, curved or arcuate, the cutting edge 22 may be defined to include other configurations, such as, for example, a linear, non-curved or non-arcuate shape. The back-relief angle θ₁₂ is configured to refract the laser beam refracted segment L₁ at the laser beam entrance face 12 according to Snell's law. The acute back-relief angle θ₁₂ of FIG. 11A (that is less than the acute back-relief angle θ₁₂ of FIG. 11B) results in the laser beam refracted segment L₁ being refracted in a direction toward the flank side face 20; however, as seen at FIG. 11B, the acute back-relief angle θ₁₂ (that is greater than the acute back-relief angle θ₁₂ of FIG. 11A) results in the laser beam refracted segment L₁ being refracted in a direction toward a secondary clearance face 34. Furthermore, in some instances, the flank face 16 of the laser-transmitting machining tool 10 f of FIG. 11A is proportionally less than the flank face16 of the laser-transmitting machining tool 10 f of FIG. 11B. Yet even further, in other examples the secondary clearance face 34 of the laser-transmitting machining tool 10 f of FIG. 11A is greater than the secondary clearance face 34 of the laser-transmitting machining tool 10 f of FIG. 11B. Accordingly, as seen at FIGS. 11A and 11B (and being similarly application to any of the laser-transmitting machining tools 10 a-10 o of the present disclosure), one or a combination of the orientation of the back-relief angle θ₁₂ and relative dimensions or lengths of any of the plurality of surfaces or faces 12-20 may change how the laser beam L travels through the laser-transmitting machining tool 10 f.

Although the laser beam entrance segment L_(E1) entering the laser beam entrance face 12 is collimated, the laser beam entrance segment L_(E1) may enter the laser beam entrance face 12 in other configurations. In some instances, the laser beam entrance segment L_(E1) may be defined by a converging laser beam (as seen in, e.g., FIGS. 5D _(C) and 5E_(C)). In other examples, the laser beam entrance segment L_(E1) may be defined by a diverging laser beam (as seen in, e.g., FIGS. 5D _(D) and 5E_(D)).

The laser-transmitting machining tools 10 f further include the secondary clearance face 34 extending between and connecting the flank face 16 to the flank side face 20. In some examples, a first end 34 ₁ of the secondary clearance face 34 extends away from the second end 20 ₂ of the flank side face 20, and a second end 34 ₂ of the secondary clearance face 34 extends away from the first end 16 ₁ of the flank face 16. In some instances, the laser-transmitting machining tool 10 f may be utilized for machining a workpiece W (see, e.g., any of FIGS. 29, and 30A-30E).

The first end 16 ₁ of the flank face 16 extends away from the second end 34 ₂ of the secondary clearance face 34 to define a flank angle or clearance angle θ₁₆. The flank angle or clearance angle θ₁₆ is obtuse.

The secondary clearance face 34 may extend away from the flank side face 20 at a secondary clearance angle θ₃₄. In some instances, the secondary clearance angle θ₃₄ is obtuse. The secondary clearance face angle θ₃₄ may be between 120° and 180°. In some implementations, the obtuse secondary clearance angle θ₃₄ is approximately equal to 150°.

As seen at FIGS. 11A and 11B, the laser beam entrance face 12 receives the laser beam entrance segment L_(E1) at a height H_(i) above the flank side face 20; the height H_(i) is a portion of a tool height H_(t) extending between the flank side face 20 and the rake side face 18. With reference to FIG. 11A, the laser beam entrance face 12 refracts the laser beam entrance segment L_(E1) to define the laser beam refracted segment L₁ that is directed toward and received by the flank side face 20. As a result of the acute back-relief angle θ₁₂ of the laser-transmitting machining tool 10 f of FIG. 11B being greater than the acute back-relief angle θ₁₂ of the laser-transmitting machining tool 10 f at FIG. 11A as described above, the laser beam entrance face 12 of the laser-transmitting machining tool 10 f of FIG. 11B refracts the laser beam entrance segment L_(E1) to define the laser beam refracted segment L₁ that is directed toward and received by the secondary clearance face 34.

With reference to FIG. 11A, the flank side face 20 receives the laser beam refracted segment L₁ at an incident mirror angle θ_(m-i) relative to the flank side face 20 (that is referenced from a reference line θ_(m) extending perpendicularly away from the flank side face 20). The flank side face 20 reflects the laser beam refracted segment L₁ to define the laser beam reflected segment L₂. The laser beam reflected segment L₂ extends away from the flank side face 20 at a reflected mirror angle θ_(m-r) (that is also referenced from the reference line θ_(m) extending perpendicularly away from the flank side face 20). The reflected mirror angle θ_(m-r) is equal to the incident mirror angle θ_(m-i). As seen at FIG. 11B, the secondary clearance face 34 receives the laser beam refracted segment L₁ at an incident mirror angle θ_(m-i) (that is referenced from a reference line θ_(m) extending perpendicularly away from the secondary clearance face 34). The secondary clearance face 34 reflects the laser beam refracted segment L₁ to define the laser beam reflected segment L₂. The laser beam reflected segment L₂ extends away from the secondary clearance face 34 at a reflected mirror angle θ_(m-r) (that is also referenced from the reference line θ_(m) extending perpendicularly away from the secondary clearance face 34). The reflected mirror angle θ_(m-r) is equal to the incident mirror angle θ_(m-i).

As seen at both of FIGS. 11A and 11B, the laser beam reflected segment L₂ is then directed toward and received by one or more of: (1) the cutting edge 22 (causing, e.g., the laser beam exit segment LE to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing, e.g., the laser beam exit segment L to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); according to an exemplary configuration of the laser-transmitting machining tools 10 f seen at FIGS. 11A and 11B, the laser beam reflected segment L₂ is then directed toward and received by the second end 14 ₂ of rake face 14 near the cutting edge 22. Thereafter, the laser beam reflected segment L₂ exits the laser-transmitting machining tools 10 f and defines the laser beam exit segment L_(E2) of the laser beam L. The laser beam exit segment L_(E2) may be refracted into the workpiece W. In some instances, the laser-transmitting machining tool 10 f is used for machining a deeply concave workpiece W, such as a workpiece W having features with a small radius of curvature (ROC) and small clear aperture (CA).

The laser beam exit segment L_(E2) exits the laser-transmitting machining tools 10 f of FIGS. 11A and 11B at a height H_(e) above the flank side face 20. In some examples, the flank side face 20 receives the laser beam reflected segment L₂ at an angle less than a critical angle θ_(C), as defined in Equation 3, when the following relationship is satisfied:

abs(θ_(m)−θ₄−θ₁₆+360°)<θ_(C),θ_(m)>θ_(C)  (46)

Although the flank side face 20 is shown reflecting the laser beam reflected segment L₂ toward the rake face 14 at FIG. 11A, in other examples, the flank side face 20 may reflect the laser beam reflected segment L₂ toward the flank face 16, causing the flank face 16 to refract the laser beam reflected segment L₂ into the workpiece W. The flank side face 20 may reflect the laser beam reflected segment L₂ toward other faces of the laser-transmitting machining tool 10 f as well. The tool length l of the laser-transmitting machining tool 10 f may be dictated by the following equation:

l=(H _(t) −H _(e))*tan(θ_(m))+(H _(t) −H _(i))*tan(θ₁₂−90°)  (47)

Referring to FIGS. 11A-12A and 11B-12B, the laser beam entrance face 12 may be defined by a substantially linear, flat or planar as the laser beam entrance face 12 that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. Although the laser beam entrance face 12 is defined by a substantially linear, flat or planar as the laser beam entrance face 12 that is substantially similar to the substantially linear, flat or planar as the laser beam entrance faces 12 of the laser-transmitting machining tools 10 a, 10 b, 10 c, and 10 d of, respectively, FIGS. 2A, 4A, 6A, and 8A, any of the other configurations described above (e.g., an inwardly-projecting axial cylindrical configuration, an outwardly-projecting axial cylindrical configuration, an inwardly-projecting lateral cylindrical configuration, an outwardly-projecting lateral cylindrical configuration, an inwardly-projecting spherical configuration, an outwardly-projecting spherical configuration, or one or more diffractive so surface portions) at FIGS. 2B-2H, 4B-4H, 6B-6H, and 8B-8H may define the laser beam entrance faces 12 of the laser-transmitting machining tools 10 f. Similarly, one or more surfaces of the plurality of surfaces 12-20 of the laser-transmitting machining tool 10 f may be partially or wholly coated with a reflection-enhancing coating 36 as similarly described above at FIGS. 2I, 4I, 6I, and 8I.

Referring now to FIGS. 13 and 14, an exemplary laser-transmitting machining tool is shown generally at 10 g. The medium of the laser-transmitting machining tool 10 g may include any desirable material such as, for example any type of single or poly crystal transmissive media including but not limited to: diamonds, sapphires, moissanites; chrysoberyls; alexandrite; and the like. In other configurations, exemplary materials defining the medium of the laser-transmitting machining tool 10 g may include but are not limited to other transmissive media such as, for example, carbides, cubic boron nitride (CBN); silicon; nitrides; steels; alloys; ceramics; alumina; glass; glass composites; composites; and the like.

Referring to FIGS. 13 and 14, the exemplary laser-transmitting machining tool 10 g is also defined a plurality of sidewall surfaces or faces 21 a-21 d The plurality of sidewall surfaces or faces 21 a-21 d includes a first upstream sidewall surface or face 21 a (see, e.g., FIGS. 13 and 14), a first downstream sidewall surface or face 21 b (see, e.g., FIGS. 13 and 14), a second upstream sidewall surface or face 21 c (see, e.g., FIG. 14), and a second downstream sidewall surface or face 21 d (see, e.g., FIG. 14). Each of the first upstream surface or face 21 a and the second upstream sidewall surface or face 21 c extends from the laser-beam entrance face 12. Each of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d extends from the rake face 14 and the flank face or clearance face 16. The first upstream surface or face 21 a meets the first downstream sidewall surface or face 21 b at a first side edge 23 a (see, e.g., FIGS. 13 and 14) that is arranged at an angle θ₂₃ (see, e.g., FIG. 13) that is substantially similar to the flank angle or clearance angle θ₁₆ that will be described in greater detail below. The first upstream surface or face 21 a and the first downstream sidewall surface or face 21 b connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 g to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tool 10 g. The second upstream sidewall surface or face 21 c meets the second downstream sidewall surface or face 21 d at a second side edge 23 b (see, e.g., FIG. 14) that is similarly arranged at the angle θ₂₃. The second upstream surface or face 21 c and the second downstream sidewall surface or face 21 d also connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 g to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tool 10 g.

The exemplary laser-transmitting machining tool 10 g is defined by a substantially similar structural configuration with respect to the transmitting machining tools 10 and 10 a-10 d of FIGS. 1A-8I and 27 described above and includes: the plurality of surfaces or faces 12-20; the first end 12 ₁-20 ₁ of each respective surface 12-20; and the second end 12 ₂-20 ₂ of each respective surface 12-20. Furthermore, the first end 14 ₁ of the rake face 14 extends away from the second end 18 ₂ of the rake side face 18 at a negative rake angle θ₁₄; accordingly, the rake face 14 may be referred to as a negative rake face. In some instances, the rake angle θ₁₄ may range between approximately 155° and an amount less than 180°

In some examples, the second end 12 ₂ of the laser-beam entrance face 12 extends away from the first end 20 ₁ of the second side face 20 at a back-relief angle θ₁₂. As seen at FIG. 13, the back-relief angle θ₁₂ is right angle (i.e., equal to 90°).

As seen at FIG. 13, the laser beam L that enters and then exits the laser-transmitting machining tool 10 g is shown being defined by a plurality of segments L_(E1), L₁, L₂, L_(E2). The plurality of segments L_(E1), L₁, L₂, L_(E2) include a laser beam entrance segment L_(E1), a laser beam segment L₁, laser beam reflected segment L₂, and a laser beam exit segment L_(E2). The laser beam entrance segment L_(E1) is collimated, which is generally defined by a tube or cylindrical arrays of rays (see, e.g., the laser beam L of FIGS. 31A-31B including a central ray Φ_(A) and circumferential arrays of rays Φ_(R1), Φ_(R2)).

With reference to FIG. 13, the laser beam entrance face 12 is configured to receive the collimated laser beam entrance segment L_(E1) such that the laser beam segment L₁ is directed toward and through one or more of: (1) the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W). The cutting edge 22 may be non-linear, curved or arcuate (as seen in, e.g., FIG. 14); although the cutting edge 22 may be non-linear, curved or arcuate, the cutting edge 22 may be defined to include other configurations, such as, for example, a linear, non-curved or non-arcuate shape. The back-relief angle θ₁₂ is configured to receive the laser beam segment L₁ at the laser beam entrance face 12 according to Snell's law. The perpendicular or right back-relief angle θ, results in the laser-transmitting machining tool 10 g not refracting the laser beam entrance segment L_(E1); rather, the laser beam entrance face 12 of the laser-transmitting machining tool 10 g permits the laser beam entrance segment L_(E1) to pass into the body of the transmitting machining tool 10 g for defining the laser beam segment L₁.

Although the laser beam entrance segment L_(E1) entering the laser beam entrance face 12 is collimated, the laser beam entrance segment L_(E1) may enter the laser beam entrance face 12 in other configurations. In some instances, the laser beam entrance segment L_(E1) may be defined by a converging laser beam (as seen in, e.g., FIGS. 5D _(C) and 5E_(C)). In other examples, the laser beam entrance segment L_(E1) may be defined by a diverging laser beam (as seen in, e.g., FIGS. 5D _(D) and 5E_(D)).

The laser-transmitting machining tool 10 g further includes a secondary clearance face 34 extending between and connecting the flank face 16 to the flank side face 20. In some examples, a first end 34 ₁ of the secondary clearance face 34 extends away from the second end 20 ₂ of the flank side face 20, and a second end 34 ₂ of the secondary clearance face 34 extends away from the first end 16 ₁ of the flank face 16. In some instances, the laser-transmitting machining tool 10 g may be utilized for machining a workpiece W (see, e.g., any of FIGS. 29, and 30A-30E).

The first end 16 ₁ of the flank face 16 extends away from the second end 34 ₂ of the secondary clearance face 34 to define a flank angle or clearance angle θ₁₆. The flank angle or clearance angle θ₁₆ is obtuse.

The secondary clearance face 34 may extend away from the flank side face 20 at a secondary clearance angle θ₃₄. In some instances, the secondary clearance angle θ₃₄ is obtuse. The secondary clearance face angle θ₃₄ may be between 120° and 180°. In some implementations, the obtuse secondary clearance angle θ₃₄ is approximately equal to 150°.

As seen at FIG. 13, the laser beam entrance face 12 receives the laser beam entrance segment L_(E1) at a height H_(i) above the flank side face 20; the height H_(i) is a portion of a tool height H_(t) extending between the flank side face 20 and the rake side face 18. The laser beam entrance face 12 receives the laser beam entrance segment L_(E1) to define the laser beam segment L₁ that is directed that is not toward either of the rake side face 18 and the flank side face 20; accordingly, the laser beam segment L₁ may be directed toward and may be received by one of: the rake face 14; the flank face 16; or the secondary clearance face 34. In some instances, the laser beam segment L₁ may be directed directly to the cutting edge 22. Any of the rake face 14, the flank face 16, the secondary clearance face 34 reflects the laser beam segment L₁ to define the laser beam reflected segment L₂; the laser beam reflected segment L₂ may then be directed toward and received by one or more of: (1) the negative rake face 14 near the cutting edge 22 (causing, e.g., the laser beam exit segment. La to refract onto the workpiece W); (2) the flank face 16 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); and (3) the secondary clearance face 34 (causing. e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W). In some instances, the laser-transmitting machining tool 10 g is used for machining a deeply concave workpiece W, such as a workpiece W having features with a small radius of curvature (ROC) and small clear aperture (CA).

In an example, the laser beam entrance face 12 receives the laser beam entrance segment LE. The secondary clearance face 34 receives the laser beam segment L₁ at an incident mirror angle θ_(m-i) (that is referenced from a reference line θ_(m) extending perpendicularly away from the secondary clearance face 34). The secondary clearance face 34 reflects the laser beam segment L₁ to define the laser beam reflected segment L₂. The laser beam reflected segment L₂ extends away from the secondary clearance face 34 at a reflected mirror angle θ_(m-r) (that is also referenced from the reference line θ_(m) extending perpendicularly away from the secondary clearance face 34). The reflected mirror angle θ_(m-r) is equal to the incident mirror angle θ_(m-i).

The laser beam reflected segment L₂ is then directed toward the and received by one of: the rake face 14, the flank face 16; and the cutting edge 22; according to an exemplary configuration of the laser-transmitting machining tool 10 g seen at FIG. 13, the laser beam reflected segment L₂ is then refracted or directed toward and received near the cutting edge 22. Thereafter, the laser beam reflected segment L₂ exits the laser-transmitting machining tool 10 g and defines the laser beam exit segment L_(E2) of the laser beam L. The laser beam exit segment L_(E2) may be refracted toward the workpiece W.

Referring to FIGS. 13 and 14, the laser beam entrance face 12 may be defined by a substantially linear, flat or planar as the laser beam entrance face 12 that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. Although the laser beam entrance face 12 is defined by a substantially linear, flat or planar as the laser beam entrance face 12 that is substantially similar to the substantially linear, flat or planar as the laser beam entrance faces 12 of the laser-transmitting machining tools 10 a, 10 b, 10 c, and 10 d of, respectively. FIGS. 2A, 4A, 6A, and 8A, any of the other configurations described above (e.g. an inwardly-projecting axial cylindrical configuration, an outwardly-projecting axial cylindrical configuration, an inwardly-projecting lateral cylindrical configuration, an outwardly-projecting lateral cylindrical configuration, an inwardly-projecting spherical configuration, an outwardly-projecting spherical configuration, or one or more diffractive surface portions) at FIGS. 2B-2H, 4B-4H, 6B-6H, and 8B-8H may define the laser beam entrance faces 12 of the laser-transmitting machining tools 10 g. Similarly, one or more surfaces of the plurality of surfaces 12-20 of the laser-transmitting machining tool 10 g may be partially or wholly coated with a reflection-enhancing coating 36 as similarly described above at FIGS. 2I, 4I, 6I, and 8I.

Referring now to FIGS. 15 and 16, an exemplary laser-transmitting machining tool is shown generally at 10 h. The medium of the laser-transmitting machining tool 10 h may include any desirable material such as, for example any type of single or poly crystal transmissive media including but not limited to: diamonds, sapphires, moissanites; chrysoberyls; alexandrite; and the like. In other configurations, exemplary materials defining the medium of the laser-transmitting machining tool 10 h may include but are not limited to other transmissive media such as, for example: carbides, cubic boron nitride (CBN); silicon; nitrides; steels; alloys; ceramics; alumina; glass; glass composites; composites; and the like.

Referring to FIGS. 15 and 16, the exemplary laser-transmitting machining tool 10 h is also defined a plurality of sidewall surfaces or faces 21 a-21 b. The plurality of sidewall surfaces or faces 21 a-21 b includes a first sidewall surface or face 21 a (see, e.g., FIGS. 15 and 16) and a second sidewall surface or face 21 b (see, e.g., FIG. 16). Each of the first sidewall surface or face 21 a and the second sidewall surface or face 21 b extends from: the laser-beam entrance face 12; the rake face 14; the flank face or clearance face 16; the flank side face 20; and the secondary clearance face 34.

The exemplary laser-transmitting machining tool 10 h is defined by a substantially similar structural configuration with respect to the transmitting machining tools 10 and 10 a-10 d of FIGS. 1A-8I and 27 described above and includes: the plurality of surfaces or faces 12-16 and 20 with the exception of a rake side face 18 (i.e., the rake face 14 of the laser-transmitting machining tool 10 h extends from and directly connects the laser beam entrance face 12 to the flank face 16). Furthermore, the first end 14 ₁ of the rake face 14 extends away from the second end 12 ₂ of the laser beam entrance face 12, and, the second end 14 ₂ of the rake face 14 extends away from the second end 16 ₂ of the flank face 16 at a rake angle θ₁₄. Unlike the exemplary laser-transmitting machining tools 10 and 10 a-10 g described above, the laser-transmitting machining tool 10 h does not define a rake angle θ₁₄ that is a negative or obtuse. As seen at FIG. 15, the second end 14 ₂ of the rake face 14 extends away from the second end 16 ₂ of the flank face 16 at a positive or acute rake angle θ₁₄; accordingly, the rake face 14 may be referred to as a positive rake face. In some instances, the positive rake angle θ₁₄ may range between approximately 70° and an amount less than 90°.

In some examples, the second end 12 ₂ of the laser-beam entrance face 12 extends away from the first end 20 ₁ of the second side face 20 at a back-relief angle θ₁₂. As seen at FIG. 15, the back-relief angle θ₁₂ is right angle (i.e, equal to 90°).

As seen at FIG. 15, the laser beam L that enters and then exits the laser-transmitting machining tool 10 h is shown being defined by a plurality of segments L_(E1), L₁, L₂, L_(E2). The plurality of segments L_(E1), L₁, L₂, L_(E2) include a laser beam entrance segment L_(E1), a laser beam segment L₁, laser beam reflected segment L₂, and a laser beam exit segment L_(E2). The laser beam entrance segment L_(E1) is collimated, which is generally defined by a tube or cylindrical arrays of rays (see, e.g., the laser beam L of FIGS. 31A-31B including a central ray Φ_(A) and circumferential arrays of rays Φ_(R1), Φ_(R2)).

With reference to FIG. 15, the laser beam entrance face 12 is configured to receive the collimated laser beam entrance segment L_(E1) such that the laser beam segment L₁ is directed toward and through one or more of: (1) the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W). The cutting edge 22 may be non-linear, curved or arcuate (as seen in, e.g., FIG. 16); although the cutting edge 22 may be non-linear, curved or arcuate, the cutting edge 22 may be defined to include other configurations, such as, for example, a linear, non-curved or non-arcuate shape. The back-relief angle θ₁₂ is configured to receive the laser beam segment L₁ at the laser beam entrance face 12 according to Snell's law. The perpendicular or right back-relief angle θ₁₂ results in the laser-transmitting machining tool 10 h not refracting the laser beam entrance segment L_(E1); rather, the laser beam entrance face 12 of the laser-transmitting machining tool 10 h permits the laser beam entrance segment L_(E1) to pass into the body of the transmitting machining tool 10 h for defining the laser beam segment L₁.

Although the laser beam entrance segment L_(E1) entering the laser beam entrance face 12 is collimated, the laser beam entrance segment L_(E1) may enter the laser beam entrance face 12 in other configurations. In some instances, the laser beam entrance segment L_(E1) may be defined by a converging laser beam (as seen in, e.g., FIGS. 5D _(C) and 5E_(C)). In other examples, the laser beam entrance segment L_(E1) may be defined by a diverging laser beam (as seen in, e.g., FIGS. 5D _(D) and 5E_(D)).

The laser-transmitting machining tool 10 h further includes a secondary clearance face 34 extending between and connecting the flank face 16 to the flank side face 20. In some examples, a first end 34 ₁ of the secondary clearance face 34 extends away from the second end 20 ₂ of the flank side face 20, and a second end 34 ₂ of the secondary clearance face 34 extends away from the first end 16 ₁ of the flank face 16. In some instances, the laser-transmitting machining tool 10 h may be utilized for machining a workpiece W (see, e.g., any of FIGS. 29, and 30A-30E).

The first end 16 ₁ of the flank face 16 extends away from the second end 34 ₂ of the secondary clearance face 34 to define a flank angle or clearance angle θ₁₆. The flank angle or clearance angle θ₁₆ is obtuse.

The secondary clearance face 34 may extend away from the flank side face 20 at a secondary clearance angle θ₃₄. In some instances, the secondary clearance angle θ₃₄ is obtuse. The secondary clearance face angle θ₃₄ may be between 120° and 180°. In some implementations, the obtuse secondary clearance angle θ₃₄ is approximately equal to 140°.

In an example, the laser beam entrance face 12 receives the laser beam entrance segment L_(E1). The laser beam entrance face 12 of the laser-transmitting machining tool 10 h does not refract the laser beam entrance segment L_(E1); rather, the laser beam entrance face 12 of the laser-transmitting machining tool 10 h permits the laser beam entrance segment L_(E1) to pass into the body of the transmitting machining tool 10 h, defining the laser beam segment L₁ that is directed toward and received by the secondary clearance face 34.

The secondary clearance face 34 receives the laser beam segment L₁ at an incident mirror angle θ_(m-i) (that is referenced from a reference line θ_(m) extending perpendicularly away from the secondary clearance face 34). The secondary clearance face 34 reflects the laser beam segment L₁ to define the laser beam reflected segment L₂. The laser beam reflected segment L₂ extends away from the secondary clearance face 34 at a reflected mirror angle θ_(m-i) (that is also referenced from the reference line θ_(m) extending perpendicularly away from the secondary clearance face 34). The reflected mirror angle θ_(m-r) is equal to the incident mirror angle θ_(m-i).

The laser beam reflected segment L₂ is then directed toward the and received by (1) the cutting edge 22 (causing. e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing. e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); according to an exemplary configuration of the laser-transmitting machining tool 10 h seen at FIG. 15, the laser beam reflected segment L₂ is then refracted or directed toward and received near the cutting edge 22. Thereafter, the laser beam reflected segment L₂ exits the laser-transmitting machining tool 10 h and defines the laser beam exit segment L_(E2) of the laser beam L. The laser beam exit segment L_(E2) may be refracted toward the workpiece W.

Referring to FIGS. 15 and 16, the laser beam entrance face 12 may be defined by a substantially linear, flat or planar as the laser beam entrance face 12 that extends between the first end 12 ₁ of the laser-beam entrance face 12 and the second end 12 ₂ of the laser-beam entrance face 12. Although the laser beam entrance face 12 is defined by a substantially linear, flat or planar as the laser beam entrance face 12 that is substantially similar to the substantially linear, flat or planar as the laser beam entrance faces 12 of the laser-transmitting machining tools 10 a, 10 b, 10 c, and 10 d of, respectively, FIGS. 2A, 4A, 6A, and 8A, any of the other configurations described above (e.g., an inwardly-projecting axial cylindrical configuration, an outwardly-projecting axial cylindrical configuration, an inwardly-projecting lateral cylindrical configuration, an outwardly-projecting lateral cylindrical configuration, an inwardly-projecting spherical configuration, an outwardly-projecting spherical configuration, or one or more diffractive surface portions) at FIGS. 2B-2H, 4B-4H, 6B-6H, and 8B-8H may define the laser beam entrance faces 12 of the law-transmitting machining tools 10 h. Similarly, one or more surfaces of the plurality of surfaces 12-20 of the laser-transmitting machining tool 10 h may be partially or wholly coated with a reflection-enhancing coating 36 as similarly described above at FIGS. 2I, 4I, 6I, and 8I.

Referring now to FIGS. 17, 17′, and 18, an exemplary laser-transmitting machining tool is shown generally at 10 i. The medium of the laser-transmitting machining tool 10 i may include any desirable material such as, for example any type of single or poly crystal transmissive media including but not limited to: diamonds; sapphires, moissanites; chrysoberyls; alexandrite; and the like. In other configurations, exemplary materials defining the medium of the laser-transmitting machining tool 10 i may include but are not limited to other transmissive media such as, for example: carbides; cubic boron nitride (CBN); silicon, nitrides, steels; alloys; ceramics, alumina; glass; glass composites; composites; and the like.

Referring to FIG. 17, the exemplary laser-transmitting machining tool 10 i is defined by a substantially similar structural configuration with respect to the transmitting machining tools 10 and 10 a-10 d of FIGS. 1A-8I and 27 described above and includes the plurality of surfaces or faces 12-20. The laser-transmitting machining tool 10 i also includes the non-linear, curved or arcuate cutting edge 22; although the cutting edge 22 may be non-linear, curved or arcuate, the cutting edge 22 may be defined to include other configurations, such as, for example, a linear, non-curved or non-arcuate shape. In some instances, the laser-transmitting machining tool 10 i may be utilized for machining a workpiece W (see. e.g., any of FIGS. 29, and 30A-30E). The laser-transmitting machining tool 10 i defines a rake angle θ₁₄ that is a negative or obtuse. The first end 16 ₁ of the flank face 16 extends away from the second end 20 ₂ of the flank side face 20 to define a flank angle or clearance angle θ₁₆. The flank angle or clearance angle θ₁₆ is obtuse.

In some examples, the second end 12 ₂ of the laser-beam entrance face 12 extends away from the first end 20 ₁ of the flank side face 20 at a back-relief angle θ₁₂. In some examples, the back-relief angle θ₁₂ is a right angle (i.e., equal to 90°). Although the back-relief angle θ₁₂ may be a right angle, the back-relief angle θ₁₂ may be obtuse (i.e., greater than 90°) or are acute (i.e., less than 90°) in a substantially similar manner as described above at, for example, FIGS. 1A-4I with respect to the laser-transmitting machining tools 10 a and 10 b.

Referring to FIGS. 17 and 18, the exemplary laser-transmitting machining tool 10 i is also defined a plurality of sidewall surfaces or faces 21 a-21 d. The plurality of sidewall surfaces or faces 21 a-21 d includes a first upstream sidewall surface or face 21 a (see, e.g., FIGS. 17 and 18), a first downstream sidewall surface or face 21 b (see, e.g., FIGS. 17 and 18), a second upstream sidewall surface or face 21 c (see, e.g., FIG. 18), and a second downstream sidewall surface or face 21 d (see, e.g., FIG. 18).

Each of the first upstream surface or face 21 a and the second upstream sidewall surface or face 21 c extends from the laser-beam entrance face 12. Each of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d extends from the rake face 14 and flank face or clearance face 16.

The first upstream surface or face 21 a meets the first downstream sidewall surface or face 21 b at a first side edge 23 a (see, e.g., FIGS. 17 and 18) that is arranged at an angle θ₂₃ (see, e.g., FIG. 17) that is substantially similar to the flank angle or clearance angle θ₁₆ that will be described in greater detail below. The first upstream surface or face 21 a and the first downstream sidewall surface or face 21 b connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 i to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tool 10 i.

The second upstream sidewall surface or face 21 c meets the second downstream sidewall surface or face 21 d at a second side edge 23 b (see. e.g., FIG. 18) that is similarly arranged at the angle θ₂₃. The second upstream surface or face 21 c and the second downstream sidewall surface or face 21 d also connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 i to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tool 10 i.

As seen at FIG. 17, the laser beam L that enters and then exits the laser-transmitting machining tools 10 i is shown being defined by a plurality of segments L_(E1), L₁, L₂, L_(E2). The plurality of segments L_(E1), L₁, L_(E2) include a laser beam entrance segment L_(E1), a laser beam refracted segment L₁, a laser beam reflected segment L₂, and a laser beam exit segment L_(E2). The laser beam entrance segment L_(E1) is collimated, which is generally defined by a tube or cylindrical arrays of rays (see, e.g., the laser beam L of FIGS. 31A-31B including a central ray Φ_(A) and circumferential arrays of rays Φ_(R1), Φ_(R2))

Although the laser beam entrance segment L_(E1) entering the laser beam entrance face 12 is collimated, the laser beam entrance segment L_(E1) may enter the laser beam entrance face 12 in other configurations. In some instances, the laser beam entrance segment L_(E1) may be defined by a converging laser beam (as seen in, e.g., FIGS. 5D _(C) and 5E_(C)). In other examples, the laser beam entrance segment L_(E1) may be defined by a diverging laser beam (as seen in, e.g., FIGS. 5D _(D) and 5E_(D)).

As seen at FIGS. 17, 17′ and 18, the laser beam entrance face 12 may be defined by a recessed, inverted or inwardly-projecting wedge shape including a first substantially linear, flat or planar laser beam entrance face segment 12 a and a second substantially linear, flat or planar laser beam entrance face segment 12 b both extending between the first end 12 ₁ of the laser-beam entrance face 12 (from the rake side face 18) and the second end 12 ₂ of the laser-beam entrance face 12 (from the flank side face 20). The first substantially linear, flat or planar laser beam entrance face segment 12 a also extends from the first upstream sidewall surface or face 21 a. The second substantially linear, flat or planar laser beam entrance face segment 12 b also extends from the second upstream sidewall surface or face 21 c. The first substantially linear, flat or planar laser beam entrance face segment 12 a and the second substantially linear, flat or planar laser beam entrance face segment 12 b meet at an entrance face edge 12 c.

With continued reference to FIG. 18, a plane P₁₂ extends across the laser-transmitting machining tool 10 i proximate the laser beam entrance face 12. In an example, the plane P₁₂ extends across an edge where: (1) the first substantially linear, flat or planar laser beam entrance face segment 12 a extends from the first upstream sidewall surface or face 21 a; and (2) the second substantially linear, flat or planar laser beam entrance face segment 12 b extends from the second upstream sidewall surface or face 21 c. Furthermore, the plane P₁₂ is substantially perpendicular with respect to both of the first upstream sidewall surface or face 21 a and the second upstream sidewall surface or face 21 c.

In some instances, each of the first substantially linear, flat or planar laser beam entrance face segment 12 a and the second substantially linear, flat or planar laser beam entrance face segment 12 b are arranged at an inwardly-projecting angle θ_(W) that defines the recessed, inverted or inwardly-projecting wedge shape of the laser beam entrance face 12 of the laser-transmitting machining tool 10 i. In some instances, the inwardly-projecting angle θ_(W) is acute (i.e., less than 90°) and projects in a direction toward the arcuate or curved cutting edge 22: in some implementations, the inwardly-projecting angle θ_(W) is approximately equal to 20°. Both of the first substantially linear, flat or planar laser beam entrance face segment 12 a and the second substantially linear, flat or planar laser beam entrance face segment 12 b are configured to refract the laser beam L at the laser beam entrance face 12 according to Snell's law.

As seen at FIG. 18, in some instances, the recessed, inverted or inwardly-projecting wedge shape of the laser beam entrance face 12 of the laser-transmitting machining tool 10 i receives the laser beam entrance segment L_(E1) of the laser beam L and thereafter causes the laser beam refracted segment L₁ of the laser beam L to become divergent after the laser beam L enters the body of the laser-transmitting machining tool 10 i. As can readily be seen in FIG. 18, the laser beam L is refracted by the entrance face 12 according to Snell's law, the refracted laser beam assuming an angle θ_(R) relative to a line R normal to the entrance face 12. θ_(R) may be expressed by the following equation:

$\begin{matrix} {\theta_{R} = {\sin^{- 1}\left( \frac{\sin \mspace{11mu} \theta_{W}}{n_{2}} \right)}} & (48) \end{matrix}$

In some examples, the recessed, inverted or inwardly-projecting wedge shape of the laser beam entrance face 12 of the laser-transmitting machining tool 10 i receives the laser beam entrance segment L_(E1) of the laser beam L and thereafter causes the laser beam refracted segment L₁ of the laser beam L to become divergent after the laser beam L enters the body of the laser-transmitting machining tool 10 i. The laser beam refracted segment L₁ of the laser beam L thereafter may be incident upon one or both of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d near the arcuate or curved cutting edge 22 for defining the laser beam reflected segment L₂. Thereafter, laser beam reflected segment L₂, of the laser beam L is reflected off one or both of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d and converges at a focal point F_(P) upstream of the arcuate or curved cutting edge 22. Thereafter, the laser beam reflected segment L₂, of the laser beam L is refracted off the arcuate or curved cutting edge 22 to define the laser beam exit segment L_(E2) that may be refracted toward the workpiece W. The angle of convergence θ_(S) of the reflected laser beam may be expressed by the following equation:

θ_(S)=180°−2(θ_(W)−θ_(R))  (49)

Referring now to FIGS. 19, 19′, and 20, an exemplary laser-transmitting machining tool is shown generally at 10 j. The medium of the laser-transmitting machining tool 10 j may include any desirable material such as, for example any type of single or poly crystal transmissive media including but not limited to: diamonds; so sapphires; moissanites, chrysoberyls, alexandrite; and the like. In other configurations, exemplary materials defining the medium of the laser-transmitting machining tool 10 j may include but are not limited to other transmissive media such as, for example: carbides; cubic boron nitride (CBN); silicon; nitrides; steels, alloys, ceramics; alumina, glass; glass composites; composites; and the like.

Referring to FIG. 19, the exemplary laser-transmitting machining tool 10 j is defined by a substantially similar structural configuration with respect to the transmitting machining tools 10 and 10 a-10 d of FIGS. 1A-8I and 27 described above and includes the plurality of surfaces or faces 12-20. The laser-transmitting machining tool 10 j also includes the non-linear, curved or arcuate cutting edge 22; although the cutting edge 22 may be non-linear, curved or arcuate, the cutting edge 22 may be defined to include other configurations, such as, for example, a linear, non-curved or non-arcuate shape. In some instances, the laser-transmitting machining tool 10 j may be utilized for machining a workpiece W (see, e.g., any of FIGS. 29, and 30A-30E). The laser-transmitting machining tool 10 j defines a rake angle θ₁₄ that is a negative or obtuse. The first end 16 ₁ of the flank face 16 extends away from the second end 20 ₂ of the flank side face 20 to define a flank angle or clearance angle θ₁₆. The flank angle or clearance angle θ₁₆ is obtuse.

In some examples, the second end 12 ₂ of the laser-beam entrance face 12 extends away from the first end 20 ₁ of the flank side face 20 at a back-relief angle θ₁₂. In some examples, the back-relief angle θ₁₂ is a right angle (i.e., equal to 90°). Although the back-relief angle θ₁₂ may be a right angle, the back-relief angle θ₁₂ may be obtuse (i.e., greater than 90°) or are acute (i.e., less than 90°) in a substantially similar manner as described above at, for example, FIGS. 1A-4I with respect to the laser-transmitting machining tools 10 a and 10 b.

Referring to FIGS. 19 and 20, the exemplary laser-transmitting machining tool 10 j is also defined a plurality of sidewall surfaces or faces 21 a-21 d. The plurality of sidewall surfaces or faces 21 a-21 d includes a first upstream sidewall surface or face 21 a (see, e.g., FIGS. 19 and 20), a first downstream sidewall surface or face 21 b (see, e.g., FIGS. 19 and 20), a second upstream sidewall surface or face 21 c (see, e.g., FIG. 20), and a second downstream sidewall surface or face 21 d (see, e.g., FIG. 20).

Each of the first upstream surface or face 21 a and the second upstream sidewall surface or face 21 c extends from the laser-beam entrance face 12. Each of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d extends from the rake face 14 and flank face or clearance face 16.

The first upstream surface or face 21 a meets the first downstream sidewall surface or face 21 b at a first side edge 23 a (see, e.g., FIGS. 19 and 20) that is arranged at an angle θ₂₃ (see, e.g., FIG. 19) that is substantially similar to the flank angle or clearance angle θ₁₆ that will be described in greater detail below. The first upstream surface or face 21 a and the first downstream sidewall surface or face 21 b connects the first side 28 (i.e, one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 j to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tool 10 j.

The second upstream sidewall surface or face 21 c meets the second downstream sidewall surface or face 21 d at a second side edge 23 b (see, e.g., FIG. 20) that is similarly arranged at the angle θ₂₃. The second upstream surface or face 21 c and the second downstream sidewall surface or face 21 d also connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 j to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tool 10 j.

As seen at FIG. 19, the laser beam L that enters and then exits the laser-transmitting machining tools 10 j is shown being defined by a plurality of segments L_(E1), L₁, L_(E2). The plurality of segments L_(E1), L₁, L_(E2) include a laser beam entrance segment L_(E1), a laser beam refracted segment L₁ and a laser beam exit segment L_(E2). The laser beam entrance segment L_(E1) is collimated, which is generally defined by a tube or cylindrical arrays of rays (see, e.g., the laser beam L of FIGS. 31A-31B including a central ray Φ_(A) and circumferential arrays of rays Φ_(R1), Φ_(R2))

Although the laser beam entrance segment L_(E1) entering the laser beam entrance face 12 is collimated, the laser beam entrance segment L_(E1) may enter the laser beam entrance face 12 in other configurations. In some instances, the laser beam entrance segment L_(E1) may be defined by a converging laser beam (as seen in, e.g., FIGS. 5D_(C) and 5E_(C)). In other examples, the laser beam entrance segment L_(E1) may be defined by a diverging laser beam (as seen in, e.g., FIGS. 5D _(D) and 5E_(D)).

As seen at FIGS. 19′ and 20, the laser beam entrance face 12 may be defined by a protruding or outwardly-projecting wedge shape including a first substantially linear, flat or planar laser beam entrance face segment 12 a and a second substantially linear, flat or planar laser beam entrance face segment 12 b both extending between the first end 12 ₁ of the laser-beam entrance face 12 (from the rake side face 18) and the second end 12 ₂ of the laser-beam entrance face 12 (from the flank side face 20). The first substantially linear, flat or planar laser beam entrance face segment 12 a also extends from the first upstream sidewall surface or face 21 a. The second substantially linear, flat or planar laser beam entrance face segment 12 b also extends from the second upstream sidewall surface or face 21 c. The first substantially linear, flat or planar laser beam entrance face segment 12 a and the second substantially linear, flat or planar laser beam entrance face segment 12 b meet at an entrance face edge 12 c.

With continued reference to FIG. 20, a plane P₁₂ extends across the laser-transmitting machining tool 10 j proximate the laser beam entrance face 12. In an example, the plane P₁₂ extends across an edge where: (1) the first substantially linear, flat or planar laser beam entrance face segment 12 a extends from the first upstream sidewall surface or face 21 a; and (2) the second substantially linear, flat or planar laser beam entrance face segment 12 b extends from the second upstream sidewall surface or face 21 c. Furthermore, the plane P₁₂ is substantially perpendicular with respect to both of the first upstream sidewall surface or face 21 a and the second upstream sidewall surface or face 21 c.

In some instances, each of the first substantially linear, flat or planar laser beam entrance face segment 12 a and the second substantially linear, flat or planar laser beam entrance face segment 12 b are arranged at an outwardly-projecting angle θ_(W)′ that defines the protruding or outwardly-projecting wedge shape of the laser beam entrance face 12 of the laser-transmitting machining tool 10 j In some instances, the outwardly-projecting angle θ_(W)′ is acute (i.e., less than 90°) and projects in a direction away from the arcuate or curved cutting edge 22, in some implementations, the outwardly-projecting angle θ_(W)′ is approximately equal to 20°. Both of the first substantially linear, flat or planar laser beam entrance face segment 12 a and the second substantially linear, flat or planar laser beam entrance face segment 12 b are configured to refract the laser beam L at the laser beam entrance face 12 according to Snell's law.

As seen at FIG. 20, in some instances, the protruding or outwardly-projecting wedge shape of the laser beam entrance face 12 of the laser-transmitting machining tool 10 j receives the laser beam entrance segment L_(E1) of the laser beam L and thereafter causes the laser beam refracted segment L₁ of the laser beam L to become convergent after the laser beam L enters the body of the laser-transmitting machining tool 10 j. As can readily be seen in FIG. 20, the laser beam L is refracted by the entrance face 12 according to Snell's law, the refracted laser beam assuming an angle θ_(R) relative to a line R normal to the entrance face 12. θ_(R) may be expressed by the following equation:

$\begin{matrix} {\theta_{R} = {\sin^{- 1}\left( \frac{\sin \mspace{11mu} \theta_{W}^{\prime}}{n_{2}} \right)}} & (50) \end{matrix}$

Accordingly, the angle of convergence θ_(S) may be expressed as a function of θ_(R) and the obtuse back-relief angle θ₁₂ by the following equation:

θ_(S)=2(θ_(W)′−θ_(R))  (51)

The laser beam refracted segment L₁ of the laser beam L thereafter converges at a focal point FP upstream of an outward-most portion of the arcuate or curved cutting edge 22. Thereafter the laser beam refracted segment L₁ diverges from the focal point F_(P) and may be incident upon the arcuate or curved cutting edge 22. Thereafter, the laser beam refracted segment L₁ of the laser beam L is refracted off the arcuate or curved cutting edge 22 to define the laser beam exit segment L_(E2).

Referring now to FIG. 21, an exemplary laser-transmitting machining tool is shown generally at 10 k The medium of the laser-transmitting machining tool 10 k may include any desirable material such as, for example any type of single or poly crystal so transmissive media including but not limited to: diamonds; sapphires; moissanites; chrysoberyls; alexandrite; and the like. In other configurations, exemplary materials defining the medium of the laser-transmitting machining tool 10 k may include but are not limited to other transmissive media such as, for example: carbides; cubic boron nitride (CBN); silicon; nitrides; steels; alloys; ceramics; alumina; glass; glass composites; composites, and the like.

The exemplary laser-transmitting machining tool 10 k is defined by a substantially similar structural configuration with respect to the transmitting machining tools 10, 10 a-10 d of FIGS. 1A-8I and 30 described above and includes the plurality of surfaces or faces 12-20 and the cutting edge 22, which may be non-linear, curved or arcuate; although the cutting edge 22 may be non-linear, curved or arcuate, the cutting edge 22 may be defined to include other configurations, such as, for example, a linear, non-curved or non-arcuate shape. In some instances, the laser-transmitting machining tool 10 k may be utilized for machining a workpiece W (see, e.g., any of FIGS. 29, and 30A-30E). Furthermore, the first end 14 ₁ of the rake face 14 extends away from the second end 18 ₂ of the rake side face 18 at a negative or obtuse rake angle θ₁₄; accordingly, the rake face 14 may be referred to as a negative rake face. The first end 16 ₁ of the flank face 16 extends away from the second end 20 ₂ of the flank side face 20 at an obtuse flank angle or clearance angle θ₁₆.

Referring to FIG. 21, the exemplary laser-transmitting machining tool 10 k is also defined a plurality of sidewall surfaces or faces 21 a-21 d. The plurality of sidewall surfaces or faces 21 a-21 d includes a first upstream sidewall surface or face 21 a, a first downstream sidewall surface or face 21 b, a second upstream sidewall surface or face 2 c (not shown/refer to, e.g., FIGS. 2A-2I above), and a second downstream sidewall surface or face 21 d (not shown/refer to, e.g., FIGS. 2A-2I above).

Each of the first upstream surface or face 21 a and the second upstream sidewall surface or face 21 c extends from the laser-beam entrance face 12. Each of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d extends from the rake face 14 and flank face or clearance face 16.

The first upstream surface or face 21 a meets the first downstream sidewall surface or face 21 b at a first side edge 23 a that is arranged at an angle θ₂₃ that is substantially similar to the flank angle or clearance angle θ₁₆ that will be described in greater detail below. The first upstream surface or face 21 a and the first downstream sidewall surface or face 21 b connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 k to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tool 10 k.

The second upstream sidewall surface or face 21 c meets the second downstream sidewall surface or face 21 d at a second side edge 23 b that is similarly arranged at the angle θ₂₃. The second upstream surface or face 21 c and the second downstream sidewall surface or face 21 d also connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 k to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tool 10 k.

In some examples, the second end 12 ₂ of the laser-beam entrance face 12 extends away from the first end 20 ₁ of the flank side face 20 at a back-relief angle θ₁₂. In some examples, the back-relief angle θ₁₂ is a right angle (i.e., equal to 90°). The laser-beam entrance face 12 is configured to receive a laser beam L. The laser beam L may be further defined by a plurality of segments L_(E1), L₁ (see. e.g., L_(1a), L_(1b)), L₂, L_(E2). The plurality of segments LEI. L_(E1), L₁ (see, e.g., L_(1a), L_(1b)), L₂, L_(E2) includes at least, for example, a laser beam entrance segment L_(E1) and a laser beam exit segment L_(E2).

As seen at FIG. 21, the laser beam L that enters and then exits the laser-transmitting machining tools 10 k is shown being defined by a plurality of segments L_(E1), L₁, L_(E2). The plurality of segments L_(E1), L₁, L_(E2), include a laser beam entrance segment L_(E1), a laser beam refracted segment L₁, a laser beam reflected segment L₂, and a laser beam exit segment L_(E2). The laser beam entrance segment L_(E1) is collimated, which is generally defined by a tube or cylindrical arrays of rays (see, e.g., the laser beam L of FIGS. 31A-31B including a central ray Φ_(A) and circumferential arrays of rays Φ_(R1), Φ_(R2)).

With reference to FIG. 21, the laser beam entrance face 12 is configured to receive and refract the collimated laser beam entrance segment L_(E1) such that the laser beam refracted segment L₁ is directed toward and through one or more of: (1) the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing. e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W). The back-relief angle θ₁₂ is configured to refract the laser beam refracted segment L₁ at the laser beam entrance face 12 according to Snell's law.

Although the laser beam entrance segment L_(E1) entering the laser beam entrance face 12 is collimated, the laser beam entrance segment L_(E1) may enter the laser beam entrance face 12 in other configurations. In some instances, the laser beam entrance segment L_(E1) may be defined by a converging laser beam (as seen in, e.g., FIGS. 5D _(C) and 5E_(C)). In other examples, the laser beam entrance segment L_(E1) may be defined by a diverging laser beam (as seen in, e.g., FIGS. 5D _(D) and 5E_(D)).

The laser beam entrance face 12 may be further defined by one or more diffractive surface portions 12 _(D). At least one of the one or more diffractive surface portions 12 _(D) of the laser beam entrance face 12 receives and diffracts the laser beam entrance segment L_(E1) of the laser beam L, splitting the laser beam refracted segment L₁ into a least a first refracted laser beam portion L_(1a) and a second refracted laser beam portion L_(1b) each having intense maxima at specific angles. The first refracted laser beam portion L₁ and the second refracted laser beam portion L₂ may diffract at a diffraction angle θ_(D). In some examples, at least one of the one or more diffractive surface portions 12 _(D) of the laser beam entrance face 12 is configured to diffract. (1) the first refracted laser beam portion L_(1a) of the laser beam refracted segment L₁ away from the flank side face 20; and (2) the second laser beam portion L_(1b) of laser beam refracted segment L₁ toward the flank side face 20.

The first refracted laser beam portion L_(1a) of the laser beam refracted segment L₁ may be directed toward one or more of: (1) the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W). The second laser beam portion L_(1b) of laser beam refracted segment L₁ reflects off of the flank side face 20 for defining the laser beam reflected segment L₂. The second laser beam portion segment L_(1b) of the laser beam refracted segment L₁ reflects off of the flanks side face 20 at an incident mirror angle θ_(m-1) (that is referenced from a reference line θ_(m) extending perpendicularly away from the flank side face 20). The second laser beam portion segment L_(1b) of the laser beam refracted segment L₁ extends away from the flank side face 20 at a reflected mirror angle θ_(m-r) (that is also referenced from the reference line θ_(m) extending perpendicularly away from the flank side face 20). The reflected mirror angle θ_(m-r) is equal to the incident mirror angle θ_(m-i). The laser beam reflected segment L₂ may be directed toward one or more of: (1) the cutting edge 22 (causing. e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W).

The first refracted laser beam portion L_(1a) of the laser beam refracted segment L₁ and the laser beam reflected segment L₂ exits the laser-transmitting machining tool 10 k at the cutting edge 22 and defines the laser beam exit segment L_(E2) of the laser beam L. The laser beam exit segment L_(E2) may be refracted into the workpiece W.

In some examples, the negative rake angle θ₁₄ and the flank angle θ₁₆ are configured to cause the rake face 14 and the flank face 16 to respectively receive each of the first laser beam portion L₁ and the second laser beam portion L₂ of laser beam L at an angle less than the critical angle θ_(c). In some examples, the negative rake angle θ₁₄ and the flank angle θ₁₆ are configured to cause the rake face 14 and the flank face 16 to refract each of the first laser beam portion L₁ and the second laser beam portion L₂ of laser beam L into the workpiece W. The first laser beam portion L₁ and the second laser beam portion L₂ of laser beam L may distribute laser power more broadly in the workpiece W than a laser beam L that is not diffracted at the laser beam entrance face 12 thereby improving the laser beam L or focus quality of the laser beam L over an effective area of the workpiece W, particularly, in some instances, when a workpiece W is defined by limited working distances.

Referring now to FIG. 22, an exemplary laser-transmitting machining tool is shown generally at 10 l. The medium of the laser-transmitting machining tool 10 l may so include any desirable material such as, for example any type of single or poly crystal transmissive media including but not limited to: diamonds; sapphires; moissanites; chrysoberyls; alexandrite; and the like. In other configurations, exemplary materials defining the medium of the laser-transmitting machining tool 10 l may include but are not limited to other transmissive media such as, for example: carbides; cubic boron nitride (CBN); silicon; nitrides; steels; alloys; ceramics; alumina; glass; glass composites; composites, and the like.

The exemplary laser-transmitting machining tool 10 l is defined by a substantially similar structural configuration with respect to the transmitting machining 10, 10 a-10 d of FIGS. 1A-8I and 30 described above and includes: the plurality of surfaces or faces 12-20; the first end 12 ₁-20 ₁ of each respective surface 12-20; and the second end 12 ₂-20 ₂ of each respective surface 12-20. Furthermore, the first end 14 ₁ of the rake face 14 extends away from the second end 18 ₂ of the rake side face 18 at a negative rake angle θ₁₄; accordingly, the rake face 14 may be referred to as a negative rake face. The first end 16 ₁ of the flank face 16 extends away from the second end 20 ₂ of the flank side face 20 to define a flank angle or clearance angle θ₁₆. The flank angle or clearance angle θ₁₆ is obtuse. In some instances, the laser-transmitting machining tool 10 l may be utilized for machining a workpiece W (see, e.g., any of FIGS. 29, and 30A-30E).

Referring to FIG. 22, the exemplary laser-transmitting machining tool 10 l is also defined a plurality of sidewall surfaces or faces 21 a-21 d. The plurality of sidewall surfaces or faces 21 a-21 d includes a first upstream sidewall surface or face 21 a, a first downstream sidewall surface or face 21 b, a second upstream sidewall surface or face 21 c (not shown/refer to, e.g., FIGS. 2A-2I above), and a second downstream sidewall surface or face 21 d (not shown/refer to, e.g., FIGS. 2A-2I above).

Each of the first upstream surface or face 21 a and the second upstream sidewall surface or face 21 c extends from the laser-beam entrance face 12. Each of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d extends from the rake face 14 and flank face or clearance face 16.

The first upstream surface or face 21 a meets the first downstream sidewall surface or face 21 b at a first side edge 23 a that is arranged at an angle θ₂₃ that is substantially similar to the flank angle or clearance angle θ₁₆ that will be described in greater detail below. The first upstream surface or face 21 a and the first downstream sidewall surface or face 21 b connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 l to the second side 30 (i.e, one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tool 10 l.

The second upstream sidewall surface or face 21 c meets the second downstream sidewall surface or face 21 d at a second side edge 23 b (not shown/refer to, e.g., FIGS. 2A-2I above) that is similarly arranged at the angle θ₂₃. The second upstream surface or face 21 c and the second downstream sidewall surface or face 21 d also connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 l to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tool 10 l.

A first end 18 ₁ of the rake side face 18 extends away from a first end 12 ₁ of the laser-beam entrance face 12. A first end 20 ₁ of the flank side face 20 extends away from a second end 12 ₂ of the laser-beam entrance face 12. A first end 14 ₁ of the rake face 14 extends away from a second end 18 ₂ of the rake side face 18. A first end 16, of the flank face 16 extends away from a second end 20 ₂ of the flank side face 20. A second end 14 ₂ of the rake face 14 is joined to a second end 16 ₂ of the flank face 16 to define a cutting edge 22 that may be non-linear, curved or arcuate; although the cutting edge 22 may be non-linear, curved or arcuate, the cutting edge 22 may be defined to include other configurations, such as, for example, a linear, non-curved or non-arcuate shape. Furthermore, the first end 14 ₁ of the rake face 14 extends away from the second end 18 ₂ of the rake side face 18 at a negative or obtuse rake angle θ₁₄; accordingly, the rake face 14 may be referred to as a negative rake face. The first end 16, of the flank face 16 extends away from the second end 20 ₂ of the flank side face 20 at an obtuse flank angle or clearance angle θ₁₆.

As seen at FIG. 22, the laser beam L that enters and then exits the laser-transmitting machining tools 10 l is shown being defined by a plurality of segments L_(E1), L₁, L_(E2). The plurality of segments L_(E1), L₁, L_(E2) include a laser beam entrance segment L_(E1), a laser beam refracted segment L₁ and a laser beam exit segment L_(E2). The laser beam entrance segment L_(E1) is collimated, which is generally defined by a tube or IV) cylindrical arrays of rays (see, e.g., the laser beam L of FIGS. 31A-31B including a central ray Φ_(A) and circumferential arrays of rays Φ_(R1), Φ_(R2)).

With reference to FIG. 22, the laser beam entrance face 12 is configured to receive and refract the collimated laser beam entrance segment L such that the laser beam refracted segment L₁ is directed toward and through one or more of: (1) the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); and (3) the Rank face 16 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W). The back-relief angle θ₁₂ is configured to refract the laser beam refracted segment L₁ at the laser beam entrance face 12 according to Snell's law.

Although the laser beam entrance segment L_(E1) entering the laser beam entrance face 12 is collimated, the laser beam entrance segment L_(E1) may enter the laser beam entrance face 12 in other configurations. In some instances, the laser beam entrance segment L_(E1) may be defined by a converging laser beam (as seen in, e.g., FIGS. 5D _(C) and 5E_(C)). In other examples, the laser beam entrance segment LEI may be defined by a diverging laser beam (as seen in, e.g., FIGS. 5D _(D) and 5E_(D)).

Unlike the laser-transmitting machining tools 10 a-10 k described above, the substantially linear laser-beam entrance face 12 of the laser-transmitting machining tool 10 l does not extend from the flank side face 20 to define a back-relief angle θ₁₂; however, the substantially linear laser-beam entrance face 12 is arranged substantially perpendicularly with respect to the flank side face 20 such that a back-relief angle is seen generally at θ₁₂, defining a right angle (i.e., θ₁₂ is equal to 90°).

Furthermore, as seen at FIG. 22, the laser-beam entrance face 12 of the laser-transmitting machining tool 10 l defines a portion of an optical lens recess 38 that is sized for receiving an optical lens 40. In some examples, the optical lens 40 may be removed or exchanged for another optical lens (not shown) having different optical properties. The optical lens 40 may be circular, oval, or the like. Furthermore, the optical lens 40 may be non-movably or fixedly disposed within the optical lens recess 38. Accordingly, so the optical lens 40 may be referred to as a non-movable or fixed optical lens.

In addition to the laser-beam entrance face 12, the optical lens recess 38 is further defined by a plurality of surface portions 38 ₁-38 ₄. In an example, a first surface portion 38 ₁ of the plurality of surface portions 38 ₁-38 ₄ defining the optical lens recess 38 extends away from the first end 12 ₁ of the laser-beam entrance face 12, and respectively, a second surface portion 38 ₂ of the plurality of surface portions 38 ₁-38 ₄ defining the optical lens recess 38 extends away from the second end 12 ₂ of the laser-beam entrance face 12. The laser-beam entrance face 12 may be substantially perpendicularly arranged with respect to both of the first surface portion 38 ₁ and the second surface portion 38 ₂. A third surface portion 38 ₃ of the plurality of surface portions 38 ₁-38 ₄ defining the optical lens recess 38 connects the first end 18 of the rake side face 18 to the first surface portion 38 ₁ of the plurality of surface portions 38 ₁-38 ₄ defining the optical lens recess 38. A fourth surface portion 38 ₄ of the plurality of surface portions 38 ₁-38 ₄ defining the optical lens recess 38 connects the first end 20 ₁ of the flank side face 20 to the second surface portion 38 ₂ of the plurality of surface portions 38 ₁-38 ₄ defining the optical lens recess 38.

As seen at FIG. 22, a first end 40 ₁ of the non-movable or fixed optical lens 40 may be disposed adjacent the first surface portion 38 ₁ of the plurality of surface portions 38 ₁-38 ₄ defining the optical lens recess 38, and respectively a second end 40 ₂ of the non-movable or fixed optical lens 40 may be disposed adjacent the second surface portion 38 ₂ of the plurality of surface portions 38 ₁-38 ₄ defining the optical lens recess 38. An upstream side 40 _(U) of the non-movable or fixed optical lens 40 is configured to receive a laser beam L and a downstream side 40 _(D) of the non-movable or fixed optical lens 40 is configured to permit the laser beam L to exit the non-movable or fixed optical lens 40 such that the laser beam L may be received by the laser beam entrance face 12. In some configurations, the downstream side 40 _(D) of the non-movable or fixed optical lens 40 is spaced apart from the laser-beam entrance face 12 to define a gap G there-between.

As seen at FIG. 22, the laser beam L that enters and then exits the laser-transmitting machining tool 10 l is shown being defined by a plurality of segments L_(E1), L₁. The plurality of segments L_(E1), L₁ include a laser beam entrance segment L_(E1) and a laser beam refracted segment L₁. The laser beam entrance segment L_(E1) is shown firstly entering the non-movable or fixed optical lens 40 at the upstream side 40 _(U) of the non-movable or fixed optical lens 40. After entering the movable or fixed optical lens 40, the laser beam L is defined by the laser beam refracted segment L₁ that starts to converge, passing through the downstream side 40 _(D) of the non-movable or fixed optical lens 40 and thereafter entering the laser-transmitting machining tool 10 l at the laser beam entrance face 12. The laser beam refracted segment L₁ of the laser beam L thereafter may be incident and converge upon a focal point F_(P) located at an outward-most portion of the arcuate or curved cutting edge 22. The focal point FP is located between the first end 16 ₁ and the second end 16 ₂ of the flank face 16.

Referring now to FIGS. 23A, 23B, and 23C, an exemplary laser-transmitting machining tool is shown generally at 10 m. The medium of the laser-transmitting machining tool 10 m may include any desirable material such as, for example any type of single or poly crystal transmissive media including but not limited to: diamonds; sapphires; moissanites; chrysoberyls; alexandrite; and the like. In other configurations, exemplary materials defining the medium of the laser-transmitting machining tool 10 m may include but are not limited to other transmissive media such as, for example: carbides, cubic boron nitride (CBN); silicon; nitrides; steels; alloys, ceramics; alumina, glass; glass composites; composites; and the like.

The exemplary laser-transmitting machining tool 10 m may be defined by a substantially similar structural configuration with respect to the transmitting machining tools 10, 10 a-10 k described above, including: a plurality of surfaces or faces 12-20; the first end 12 ₁-20 ₁ of each respective surface 12-20; and the second end 12 ₂-20 ₂ of each respective surface 12-20. Furthermore, the first end 14 ₁ of the rake face 14 extends away from the second end 18 ₂ of the rake side face 18 at a negative rake angle θ₁₄; accordingly, the rake face 14 may be referred to as a negative rake face. The first end 16 ₁ of the flank face 16 extends away from the second end 20 ₂ of the flank side face 20 at an obtuse flank angle or clearance angle θ₁₆. The laser-transmitting machining tool 10 m is configured to machine a workpiece W (see, e.g., any of FIGS. 29, and 30A-30E).

Referring to FIGS. 23A-23C, the exemplary laser-transmitting machining tool 10 m is also defined a plurality of sidewall surfaces or faces 21 a-21 d. The plurality of sidewall surfaces or faces 21 a-21 d includes a first upstream sidewall surface or face 21 a, a first downstream sidewall surface or face 21 b, a second upstream sidewall surface or face 21 c (not shown/refer to, e.g., FIGS. 2A-2I above), and a second downstream sidewall surface or face 21 d (not shown/refer to, e.g., FIGS. 2A-2I above).

Each of the first upstream surface or face 21 a and the second upstream sidewall surface or face 21 c extends from the laser-beam entrance face 12. Each of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d extends from the rake face 14 and flank face or clearance face 16.

The first upstream surface or face 21 a meets the first downstream sidewall surface or face 21 b at a first side edge 23 a that is arranged at an angle θ₂₃ that is substantially similar to the flank angle or clearance angle θ₁₆ that will be described in greater detail below. The first upstream surface or face 21 a and the first downstream sidewall surface or face 21 b connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 m to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tool 10 m.

The second upstream sidewall surface or face 21 c meets the second downstream sidewall surface or face 21 d at a second side edge 23 b (not shown/refer to, e.g., FIGS. 2A-2I above) that is similarly arranged at the angle θ₂₃. The second upstream surface or face 21 c and the second downstream sidewall surface or face 21 d also connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 m to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tool 10 m.

A first end 18 ₁ of the rake side face 18 extends away from a first end 12 ₁ of the laser-beam entrance face 12. A first end 20 ₁ of the flank side face 20 extends away from a second end 12 ₂ of the laser-beam entrance face 12. A first end 14 ₁ of the rake face 14 extends away from a second end 18 ₂ of the rake side face 18. A first end 16 ₁ of the flank face 16 extends away from a second end 20 ₂ of the flank side face 20. A second end 14 ₂ of the rake face 14 is joined to a second end 16 ₂ of the flank face 16 to define a cutting edge 22 that may be non-linear, curved or arcuate, although the cutting edge 22 may be non-linear, curved or arcuate, the cutting edge 22 may be defined to include other configurations, such as, for example, a linear, non-curved or non-arcuate shape. Furthermore, the first end 14 ₁ of the rake face 14 extends away from the second end 18 ₂ of the rake side face 18 at a negative or obtuse rake angle θ₁₄; accordingly, the rake face 14 may be referred to as a negative rake face. The first end 16 ₁ of the flank face 16 extends away from the second end 20 ₂ of the flank side face 20 at an obtuse flank angle or clearance angle θ₁₆.

As seen at FIGS. 23A-23C, the laser beam L that enters and then exits the laser-transmitting machining tools 10 m is shown being defined by a plurality of segments L_(E1), L₁, L_(E2). The plurality of segments L_(E1), L₁, L_(E2) include a laser beam entrance segment L_(E1), a laser beam refracted segment L₁ and a laser beam exit segment L_(E2). The laser beam entrance segment L_(E1) is collimated, which is generally defined by a tube or cylindrical arrays of rays (see, e.g., the laser beam L of FIGS. 31A-31B including a central ray Φ_(A) and circumferential arrays of rays Φ_(R1), Φ_(R2)).

With reference to FIGS. 23A-23C, the laser beam entrance face 12 is configured to receive and refract the collimated laser beam entrance segment L_(E1) such that the laser beam refracted segment L₁ is directed toward and through one or more of: (1) the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W), (2) the negative rake face 14 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W). The back-relief angle θ₁₂ is configured to refract the laser beam refracted segment L₁ at the laser beam entrance face 12 according to Snell's law.

Although the laser beam entrance segment L_(E1) entering the laser beam entrance face 12 is collimated, the laser beam entrance segment L_(E1) may enter the laser beam entrance face 12 in other configurations. In some instances, the laser beam entrance segment L_(E1) may be defined by a converging laser beam (as seen in, e.g., FIGS. 5D _(C) and 5E_(C)). In other examples, the laser beam entrance segment L_(E1) may be defined by a diverging laser beam (as seen in, e.g., FIGS. 5D _(D) and 5E_(D)).

Unlike the laser-transmitting machining tools 10, 10 a-10 k described above, the laser-transmitting machining tool 10 m does not include a substantially linear laser-beam entrance face 12 that cooperates with the flank side face 20 to define a back-relief angle θ₁₂. Rather, the laser beam entrance face 12 may be defined by a sinusoidal (e.g., a third-order polynomial) surface having a non-linear, arcuate, or curved configuration extending between the first upstream sidewall surface or face 21 a and the second upstream sidewall surface or face 21 c.

The optical shape of the non-linear, arcuate, or curved laser-beam entrance face 12 may be defined by a third-order polynomial, using an x-y coordinate system with the origin at the center of the optical shape. For example, the thickness Z of the optical shape may be defined by the following equation, where H is a scale factor.

Z(x,y)=H(x ³+3xy ²)  (52)

Furthermore, the laser-transmitting machining tool 10 m may be arranged proximate an optical lens system 42 including an optical lens 44 and a movement actuator 46 connected to the optical lens 44. The optical lens 44 may be made from the same material as the laser-transmitting machining tool 10 m. As will be described in the following disclosure, the movement actuator 46 causes the optical lens 44 to be moveably arranged relative to the laser-transmitting machining tool 10 m. Accordingly, the optical lens 44 may be referred to as a movable or non-fixed optical lens. The movement actuator 46 may cause the moveable or non-fixed optical lens to move substantially perpendicularly with respect to a central axis of the laser beam L.

The movable or non-fixed optical lens 44 may have the same index of refraction n₂ as the laser-transmitting machining tool 10 m. In some examples, the movable or non-fixed optical lens 44 is separated from the entrance face 12 by a gap G or a material having a different index of refraction (e.g., air n₁). The movement actuator 46 imparts movement to the movable or non-fixed optical lens 44 relative to the laser beam entrance face 12 in order to achieve different optical properties of the laser-transmitting machining tool 10 m. As will be described in the following disclosure, the arrangement of the movable or non-fixed optical lens 44 relative the laser-transmitting machining tool 10 m of FIG. 23A shows the laser beam L becoming generally diverging; in other words, the arrangement of the movable or non-fixed optical lens 44 relative the laser-transmitting machining tool 10 m shown at FIG. 23A provides a defocusing optical so property. The arrangement of the movable or non-fixed optical lens 44 relative the laser-transmitting machining tool 10 m shown at FIG. 23B neither focuses nor defocuses the laser beam L. The arrangement of the movable or non-fixed optical lens 44 relative the laser-transmitting machining tool 10 m shown at FIG. 23C shows the laser beam L becoming generally converging; in other words, the arrangement of the movable or non-fixed optical lens 44 relative the laser-transmitting machining tool 10 m shown at FIG. 23C provides a focusing optical property.

In some examples, the movable or non-fixed optical lens 44 and the laser beam entrance face 12 may form a lens system with a focal length l_(f) inversely proportional to a lateral shift δ. Assuming n₁=1 for air, the focal length l_(f) may be approximately dictated by the following equation.

l _(f)=1/(6Hδ(n ₂−1))  (53)

As seen at FIGS. 23A-23C, a first end 44 ₁ of the movable or non-fixed optical lens 44 may be connected to the movement actuator 46. An upstream side 44 _(U) of the movable or non-fixed optical lens 44 is configured to receive a laser beam L and a downstream side 44 _(D) of the movable or non-fixed optical lens 44 is configured to permit the laser beam L to exit the movable or non-fixed optical lens 44 such that the laser beam L may be received by the laser beam entrance face 12. In some configurations, the downstream side 44 _(D) of the movable or non-fixed optical lens 44 is spaced apart from the laser-beam entrance face 12 to define the gap G there-between.

The upstream side 44 _(U) of the movable or non-fixed optical lens 44 may be substantially perpendicular to both of the first end 44 ₁ and the second end 44 ₂ of the movable or non-fixed optical lens 44. Furthermore, the downstream side 44 _(D) of the movable or non-fixed optical lens 44 may be defined by a sinusoidal (e.g., a third-order polynomial) surface having a non-linear, arcuate or curved configuration that is similar to the laser-beam entrance face 12; accordingly, as seen at FIG. 23B, when the first end 44 ₁ and the second end 44 ₂ of the movable or non-fixed optical lens 44 are respectively aligned with the second upstream sidewall surface or face 21 c and the first upstream sidewall surface or face 21 a of the laser-transmitting machining tool 10 m, a distance so between downstream side 44 _(D) of the movable or non-fixed optical lens 44 and the laser beam entrance face 12 of the laser-transmitting machining tool 10 m defining the gap G is the same along a width of the non-fixed optical lens 44 and the laser-transmitting machining tool 10 m. Conversely, as seen at FIGS. 23A and 23C, when the first end 44 ₁ and the second end 44 ₂ of the movable or non-fixed optical lens 44 are not respectively aligned with the second upstream sidewall surface or face 21 c and the first upstream sidewall surface or face 21 a of the laser-transmitting machining tool 10 m, the distance between the downstream side 44 _(D) of the movable or non-fixed optical lens 44 and the laser beam entrance face 12 of the laser-transmitting machining tool 10 m defining the gap G is not the same along a width of the non-fixed optical lens 44 and the laser-transmitting machining tool 10 m.

As seen at FIGS. 23A-23C, the laser-beam entrance face 12 is configured to receive a laser beam L. The laser beam L may be further defined by a plurality of segments L_(E1), L₁, L_(E2). The plurality of segments L_(E1), L₁, L_(E2) includes at least, for example, a laser beam entrance segment L_(E1), a laser beam refracted segment L₁ and a laser beam exit segment L_(E2).

The laser beam entrance segment L_(E1) is shown firstly entering the upstream side 44 _(U) of the movable or non-fixed optical lens 44 and then subsequently exits the downstream side 44 _(D) of the movable or non-fixed optical lens 44. The laser beam L then enters the gap G and is subsequently incident upon the laser beam entrance face 12 of the laser-transmitting machining tool 10 m.

Referring initially to FIG. 23A, the movement actuator 46 arranges the movable or non-fixed optical lens 44 in an first orientation whereby the gap G between the downstream side 44 _(D) of the movable or non-fixed optical lens 44 and the laser beam entrance face 12 of the laser-transmitting machining tool 10 m is relatively smaller (when compared to the orientation of FIG. 23C). As a result of the orientation of movable or non-fixed optical lens 44 relative the laser beam entrance face 12, the laser beam entrance face 12 receives the laser beam entrance segment L_(E1) of the laser beam L and thereafter causes the laser beam refracted segment L₁ of the laser beam L to become divergent after the laser beam L enters the body of the laser-transmitting machining tool 10 m. The laser beam refracted segment L₁ of the laser beam L thereafter may be incident upon one or more of the rake face 14, the flank face 16, and the arcuate or curved cutting edge 22. Thereafter, the laser beam refracted segment L₁ of the laser beam L may be refracted off one or more of the rake face 14, the flank face 16, and the arcuate or curved cutting edge 22 to define the laser beam exit segment L_(E2). The laser beam exit segment L_(E2) converges upon a focal point F_(P) located away from an outwardly most portion of the arcuate or curved cutting edge 22 at a first distance D1.

Referring to FIG. 23B, the movement actuator 46 arranges the movable or non-fixed optical lens 44 in a second or “neutral” orientation whereby the gap G between the downstream side 44 _(D) of the movable or non-fixed optical lens 44 and the laser beam entrance face 12 of the laser-transmitting machining tool 10 m is the same along a width of the non-fixed optical lens 44 and the laser-transmitting machining tool 10 m. As a result of the orientation of movable or non-fixed optical lens 44 relative the laser beam entrance face 12, the laser beam entrance face 12 receives the laser beam entrance segment LE of the laser beam L and thereafter does not change a direction of the laser beam defined by the laser beam entrance segment L_(E1) (i.e., the orientation of the non-fixed optical lens 44 relative the laser-transmitting machining tool 10 m does not cause the laser beam refracted segment L₁ of the laser beam L to become divergent or convergent after the laser beam L enters the body of the laser-transmitting machining tool 10 m; as a result, the laser beam L may remain collimated). The laser beam refracted segment L₁ of the laser beam L thereafter may be incident upon one or more of the rake face 14, the flank face 16, and the arcuate or curved cutting edge 22. Thereafter, the laser beam refracted segment L₁ of the laser beam L may be refracted off one or more of the rake face 14, the flank face 16, and the arcuate or curved cutting edge 22 to define the laser beam exit segment L_(E2). The laser beam exit segment L_(E2) converges upon a focal point F_(P) located away from an outwardly most portion of the arcuate or curved cutting edge 22 at a first distance D1.

Referring to FIG. 23C, the movement actuator 46 arranges the movable or non-fixed optical lens 44 in a third orientation whereby the gap G between the downstream side 44 _(D) of the movable or non-fixed optical lens 44 and the laser beam entrance face 12 of the laser-transmitting machining tool 10 m is relatively larger (when compared to the orientation of FIG. 23A). As a result of the orientation of movable or non-fixed optical lens 44 relative the laser beam entrance face 12, the laser beam entrance face 12 receives the laser beam entrance segment L_(E1) of the laser beam L and thereafter causes the laser beam refracted segment L₁ of the laser beam L to become convergent after the laser beam L enters the body of the laser-transmitting machining tool 10 m. The laser beam refracted segment L₁ of the laser beam L thereafter may be incident upon a focal point F_(P) located at the arcuate or curved cutting edge 22. Thereafter, the laser beam refracted segment L₁ of the laser beam L is refracted off the arcuate or curved cutting edge 22 to define the laser beam exit segment L_(E2). The laser beam exit segment L_(E2) diverges from the focal point F_(P).

Referring now to FIGS. 24A, 24B, and 24C, an exemplary laser-transmitting machining tool is shown generally at 10 n. The medium of the laser-transmitting machining tool 10 n may include any desirable material such as, for example any type of single or poly crystal transmissive media including but not limited to: diamonds; sapphires, moissanites; chrysoberyls; alexandrite; and the like. In other configurations, exemplary materials defining the medium of the laser-transmitting machining tool 10 n may include but are not limited to other transmissive media such as, for example: carbides, cubic boron nitride (CBN); silicon; nitrides; steels; alloys; ceramics; alumina; glass; glass composites; composites; and the like.

The exemplary laser-transmitting machining tool 10 n is defined by a substantially similar structural configuration with respect to the transmitting machining tools 10, 10 a-10 d of FIGS. 1A-8I and 30 described above, including: a plurality of surfaces or faces 12-20; the first end 12 ₁-20 ₁ of each respective surface 12-20; and the second end 12 ₂-20 ₂ of each respective surface 12-20. Furthermore, the first end 14 ₁ of the rake face 14 extends away from the second end 18 ₂ of the rake side face 18 at a negative rake angle θ₁₄; accordingly, the rake face 14 may be referred to as a negative rake face. The first end 16 ₁ of the flank face 16 extends away from the second end 20 ₂ of the flank side face 20 at an obtuse flank angle or clearance angle θ₁₆. The laser-transmitting machining tool 10 n is configured to machine a workpiece W (see, e.g., any of FIGS. 29, and 30A-30E).

In some examples, the second end 12 ₂ of the laser-beam entrance face 12 extends away from the first end 20 ₁ of the second side face 20 at a back-relief angle θ₁₂. In some configurations, the back-relief angle θ₁₂ may be a right angle (i.e, equal to 90°). In other configurations, the back-relief angle θ₁₂ may be obtuse (i.e., greater than 90°). In yet other configurations, the back-relief angle θ₁₂ that is acute (i.e., less than 90°).

Referring to FIGS. 24A-24C, the exemplary laser-transmitting machining tool 10 n is also defined a plurality of sidewall surfaces or faces 21 a-21 d. The plurality of sidewall surfaces or faces 21 a-21 d includes a first upstream sidewall surface or face 21 a, a first downstream sidewall surface or face 21 b, a second upstream sidewall surface or face 21 c (not shown/refer to, e.g., FIGS. 2A-2I above), and a second downstream sidewall surface or face 21 d (not shown/refer to, e.g., FIGS. 2A-2I above).

Each of the first upstream surface or face 21 a and the second upstream sidewall surface or face 21 c extends from the laser-beam entrance face 12. Each of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d extends from the rake face 14 and flank face or clearance face 16.

The first upstream surface or face 21 a meets the first downstream sidewall surface or face 21 b at a first side edge 23 a that is arranged at an angle θ₂₃ that is substantially similar to the flank angle or clearance angle θ₁₆ that will be described in greater detail below. The first upstream surface or face 21 a and the first downstream sidewall surface or face 21 b connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 n to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tool 10 n.

The second upstream sidewall surface or face 21 c meets the second downstream sidewall surface or face 21 d at a second side edge 23 b (not shown/refer to, e.g., FIGS. 2A-2I above) that is similarly arranged at the angle θ₂₃. The second upstream surface or face 21 c and the second downstream sidewall surface or face 21 d also connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 n to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tool 10 n.

A first end 18 ₁ of the rake side face 18 extends away from a first end 12 ₁ of the laser-beam entrance face 12. A first end 20 ₁ of the flank side face 20 extends away from a second end 12 ₂ of the laser-beam entrance face 12. A first end 14 ₁ of the rake face 14 extends away from a second end 18 ₂ of the rake side face 18. A first end 16 ₁ of the flank face 16 extends away from a second end 20 ₂ of the flank side face 20. A second end 14 ₂ of the rake face 14 is joined to a second end 16 ₂ of the flank face 16 to define a cutting edge 22 that may be non-linear, curved or arcuate; although the cutting edge 22 may be non-linear, curved or arcuate, the cutting edge 22 may be defined to include other configurations, such as, for example, a linear, non-curved or non-arcuate shape. Furthermore, the first end 14 ₁ of the rake face 14 extends away from the second end 18 ₂ of the rake side face 18 at a negative or obtuse rake angle θ₁₄; accordingly, the rake face 14 may be referred to as a negative rake face. The first end 16 ₁ of the flank face 16 extends away from the second end 20 ₂ of the flank side face 20 at an obtuse flank angle or clearance angle θ₁₆.

As seen at FIGS. 24A-24C, the laser beam L that enters and then exits the laser-transmitting machining tools 10 n is shown being defined by a plurality of segments L_(E1), L₁, L_(E2). The plurality of segments L_(E1), L₁, L_(E2) include a laser beam entrance segment L_(E1), a laser beam refracted segment L₁ and a laser beam exit segment L_(E2). The laser beam entrance segment L_(E1) is collimated, which is generally defined by a tube or cylindrical arrays of rays (see, e.g., the laser beam L of FIGS. 34A-34B including a central ray Φ_(A) and circumferential arrays of rays Φ_(R1). Φ_(R2)).

With reference to FIGS. 24A-24C, the laser beam entrance face 12 is configured to receive and refract the collimated laser beam entrance segment L_(E1) such that the laser beam refracted segment L₁ is directed toward and through one or more of: (1) the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); (2) the negative rake face 14 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W); and (3) the flank face 16 near the cutting edge 22 (causing, e.g., the laser beam exit segment L_(E2) to refract onto the workpiece W). The back-relief angle θ₁₂ is configured to refract the laser beam refracted segment L₁ at the laser beam entrance face 12 according to Snell's law.

Although the laser beam entrance segment L_(E1) entering the laser beam entrance face 12 is collimated, the laser beam entrance segment L_(E1) may enter the laser beam entrance face 12 in other configurations. In some instances, the laser beam entrance segment L_(E1) may be defined by a converging laser beam (as seen in, e.g., FIGS. 5D_(C) and 5E_(C)). In other examples, the laser beam entrance segment L_(E1) may be defined by a diverging laser beam (as seen in, e.g., FIGS. 5D _(D) and 5E_(D)).

Furthermore, the laser-transmitting machining tool 10 n may be arranged proximate an optical prism system 48 including a first right angle prism 50, a second right angle prism 52 and a movement actuator 54 connected to the second right angle prism 52. As will be described in the following disclosure, the movement actuator 54 causes the second right angle prism 52 to be moveably-arranged relative to a non-movable, fixed or grounded orientation of the right angle prism 50 and the laser-transmitting machining tool 10 n. Accordingly, the first right angle prism 50 may be referred to as a non-movable or fixed right angle prism the second right angle prism 52 may be referred to as a movable or non-fixed right angle prism. The movement actuator 54 may cause the moveable or non-fixed right angle prism 52 to move in a similar axial direction as that of the laser beam L.

The first, non-movable or fixed right angle prism 50 is defined by an acute prism angle θ_(P). The second, moveable or non-fixed right angle prism 52 is also defined by the same acute prism angle θ_(P). The first, non-movable or fixed right angle prism 50 is positioned to firstly receive the laser beam L and then subsequently transmit the laser beam L through the second, moveable or non-fixed right angle prism 52 toward the laser beam entrance face 12.

In some examples, the first, non-movable or fixed right angle prism 50 is defined by a prism index of refraction n and the second, moveable or non-fixed right angle prism 52 is also defined by the same prism index of refraction n_(p). In some examples, the first, non-movable or fixed right angle prism 50 and the second, moveable or non-fixed right angle prism 52 are separated by a separation distance h by a material, such as, for example, air that is defined by a different index of refraction (e.g., n₁). The laser beam L is subsequently received by the laser beam entrance face 12 after being laterally shifted (see, e.g., L_(shift) in equation 43) by the first, non-movable or fixed right angle prism 50 and the second, moveable or non-fixed right angle prism 52. The lateral shift L_(shift) may be proportional to a separation distance change (see, e.g., Δh) according to so the following equation:

$\begin{matrix} {L_{shift} = {\frac{n_{p}{\sin \left( {2\theta_{p}} \right)}}{2\sqrt{n_{1}^{2} - {n_{p}^{2}{\sin \left( \theta_{p} \right)}}}}\Delta \; h}} & (54) \end{matrix}$

The separation distance h between the first, non-movable or fixed right angle prism 50 and the second, moveable or non-fixed right angle prism 52 adjustable as a result of the movement actuator 54 imparting movement to the second, moveable or non-fixed right angle prism 52. With reference to FIG. 24A, the first, non-movable or fixed right angle prism 50 is separated from the second, moveable or non-fixed right angle prism 52 by a separation distance h that is less than a second separation distance h seen at FIGS. 24B and 24C. As seen at FIG. 24B, the first, non-movable or fixed right angle prism 50 is separated from the second, moveable or non-fixed right angle prism 52 by a separation distance h that is greater than the first separation distance h of FIG. 24A but is less than a third separation distance h of FIG. 24C. Accordingly, depending on the selected orientation of the second, moveable or non-fixed right angle prism 52 relative the first, non-movable or fixed right angle prism 50, the axial orientation of the laser beam L is selectively adjustable at the entrance face 12 such that the laser beam may exit the laser-transmitting machining tool 10 n at any of the rake face 14, the flank face16 or the arcuate or curved cutting edge 22.

Referring now to FIGS. 25, 25′, 26, and 26′ an exemplary laser-transmitting machining tool is shown generally at 10 o. The medium of the laser-transmitting machining tool 10 o may include any desirable material such as, for example any type of single or poly crystal transmissive media including but not limited to: diamonds; sapphires; moissanites, chrysoberyls, alexandrite; and the like. In other configurations, exemplary materials defining the medium of the laser-transmitting machining tool 10 o may include but are not limited to other transmissive media such as, for example: carbides; cubic boron nitride (CBN); silicon; nitrides; steels, alloys, ceramics; alumina, glass; glass composites; composites; and the like.

Referring to FIG. 25, the exemplary laser-transmitting machining tool 10 o is defined by a substantially similar structural configuration with respect to the transmitting machining tools 10 and 10 a-10 d of FIGS. 1A-81 and 27 described above and includes the plurality of surfaces or faces 12-20. The laser-transmitting machining tool 10 o also includes a “hybrid” or “split radius” cutting edge 22. Unlike the exemplary embodiments described above, as seen at FIG. 26′ the hybrid or split radius cutting edge 22 may not be limited to being defined by non-linear, curved, or arcuate configuration (or, alternatively, a linear, non-curved, or non-arcuate configuration), but, rather, the hybrid or split radius cutting edge 22 may be defined to include a combination of: (1) a non-linear, curved, or arcuate portion (see, e.g., portion of the cutting edge 22 at reference numeral 21 d 2), and (2) a linear, non-curved, or non-arcuate portion (see, e.g., portion of the cutting edge 22 at reference numeral 21 b ₂). In some instances, the laser-transmitting machining tool 10 o may be utilized for machining a workpiece W (see, e.g., any of FIGS. 29, and 30A-30E). The laser-transmitting machining tool 10 o defines a rake angle θ₁₄ that is a negative or obtuse. The first end 16 ₁ of the flank face 16 extends away from the second end 20 ₂ of the flank side face 20 to define a flank angle or clearance angle θ₁₆. The flank angle or clearance angle θ₁₆ is obtuse.

In some examples, the second end 12 ₂ of the laser-beam entrance face 12 extends away from the first end 20 ₁ of the flank side face 20 at a back-relief angle θ₁₂. In some examples, the back-relief angle θ₁₂ is a right angle (i.e., equal to 90°). Although the back-relief angle θ₁₂ may be a right angle, the back-relief angle θ₁₂ may be obtuse (i.e., greater than 90°) or are acute (i.e., less than 90°) in a substantially similar manner as described above at, for example, FIGS. 1A-4I with respect to the laser-transmitting machining tools 10 a and 10 b.

Referring to FIGS. 25 and 26, the exemplary laser-transmitting machining tool 10 o is also defined a plurality of sidewall surfaces or faces 21 a-21 d. The plurality of sidewall surfaces or faces 21 a-21 d includes a first upstream sidewall surface or face 21 a (see, e.g., FIGS. 25 and 26), a first downstream sidewall surface or face 21 b (see, e.g., FIGS. 19 and 20), a second upstream sidewall surface or face 21 c (see, e.g., FIG. 20), and a second downstream sidewall surface or face 21 d (see, e.g., FIG. 20).

Each of the first upstream surface or face 21 a and the second upstream sidewall surface or face 21 c extends from the laser-beam entrance face 12. Each of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d extends from the rake face 14 and flank face or clearance face 16.

The first upstream surface or face 21 a meets the first downstream sidewall surface or face 21 b at a first side edge 23 a (see, e.g., FIGS. 25 and 26) that is arranged at an angle θ₂₃ (see, e.g., FIG. 25) that is substantially similar to the flank angle or clearance angle θ₁₆ that will be described in greater detail below. The first upstream surface or face 21 a and the first downstream sidewall surface or face 21 b connects the first side 28 (i.e, one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 o to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tool 10 o.

The second upstream sidewall surface or face 21 c meets the second downstream sidewall surface or face 21 d at a second side edge 23 b (see, e.g., FIG. 26) that is similarly arranged at the angle θ₂₃. The second upstream surface or face 21 c and the second downstream sidewall surface or face 21 d also connects the first side 28 (i.e., one or both of the rake face 14 and first side face 18) of the laser-transmitting machining tool 10 o to the second side 30 (i.e., one or both of the flank face16 and the second side face 20) of the laser-transmitting machining tool 10 o.

As seen at FIG. 26′, the first downstream sidewall surface or face 21 b is defined by a first portion 21 b ₁ and a second portion 21 b ₂, and the second downstream sidewall surface or face 21 d is defined by a first portion 21 d ₁ and a second portion 21 d ₂. The first portion 21 b ₁, 21 d ₁ of both of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d extends along the rake side face18. The second portion 21 b ₂, 21 d ₂ of both of the first downstream sidewall surface or face 21 b and the second downstream sidewall surface or face 21 d extends along the rake face14. The second portion 21 b ₂ of the first downstream sidewall surface or face 21 b define a linear, non-curved, or non-arcuate portion of the hybrid or split radius cutting edge 22 whereas the second portion 21 d ₂ of the second downstream sidewall surface or face 21 b defines a non-linear, curved, or arcuate portion of the hybrid or split radius cutting edge 22.

As seen at FIG. 25, the laser beam L that enters and then exits the laser-transmitting machining tools 10 o is shown being defined by a plurality of segments L_(E1), L₁, L_(E2). The plurality of segments L_(E1), L₁, L_(E2) include a laser beam entrance segment L_(E1), a laser beam refracted segment L₁ and a laser beam exit segment L_(E2). The laser beam entrance segment L_(E1) is collimated, which is generally defined by a tube or cylindrical arrays of rays (see, e.g., the laser beam L of FIGS. 31A-31B including a central ray Φ_(A) and circumferential arrays of rays Φ_(R1), Φ_(R2)).

Although the laser beam entrance segment L_(E1) entering the laser beam entrance face 12 is collimated, the laser beam entrance segment L_(E1) may enter the laser beam entrance face 12 in other configurations. In some instances, the laser beam entrance segment L_(E1) may be defined by a converging laser beam (as seen in, e.g., FIGS. 5D _(C) and 5E_(C)). In other examples, the laser beam entrance segment L_(E1) may be defined by a diverging laser beam (as seen in, e.g., FIGS. 5D _(D) and 5E_(D)).

As seen at FIGS. 25′ and 26, the laser beam entrance face 12 may be defined by a protruding or outwardly-projecting partial wedge shape including a first substantially linear, flat or planar laser beam entrance face segment 12 a and a second substantially linear, flat or planar laser beam entrance face segment 12 b both extending between the first end 12 ₁ of the laser-beam entrance face 12 (from the rake side face 18) and the second end 12 ₂ of the law-beam entrance face 12 (from the flank side face 20). The first substantially linear, flat or planar laser beam entrance face segment 12 a also extends from the first upstream sidewall surface or face 21 a. The second substantially linear, flat or planar laser beam entrance face segment 12 b also extends from the second upstream sidewall surface or face 21 c The first substantially linear, flat or planar laser beam entrance face segment 12 a and the second substantially linear, flat or planar laser beam entrance face segment 12 b meet at an entrance face edge 12 c. As will be described in the following disclosure, the first substantially linear, flat or planar laser beam entrance face segment 12 a may be alternatively referred to as a substantially linear, flat or planar laser beam function entrance face segment, and the second substantially linear, flat or planar laser beam entrance face segment 12 b may be alternatively referred to as a substantially linear, flat or planar laser beam non-functional entrance face segment.

With continued reference to FIG. 26, a plane P₁₂ extends across the laser-transmitting machining tool 10 o proximate the laser beam entrance face 12. In an example, the plane P₁₂ extends across an edge where the substantially linear, flat or planar laser beam functional entrance face segment 12 a extends from the first upstream sidewall surface or face 21 a. Furthermore, the plane P₁₂ is substantially perpendicular with respect to the first upstream sidewall surface or face 21 a. The substantially linear, flat or planar laser beam non-functional entrance face segment 12 b perpendicularly extends from the second downstream sidewall surface or face 21 d and meets the substantially linear, flat or planar laser beam functional entrance face segment 12 a at the entrance face edge 12 c. The substantially linear, flat or planar laser beam non-functional entrance face segment 12 b may be spaced apart from the plane P₁₂ at a distance d.

In some instances, the substantially linear, flat or planar laser beam functional entrance face segment 12 a is arranged at an outwardly-projecting angle θ_(W)′ that defines the protruding or outwardly-projecting partial wedge shape of the laser beam entrance face 12 of the laser-transmitting machining tool 10 o. In some instances, the outwardly-projecting angle θ_(W)′ is acute (i.e., less than 90°) and projects in a direction away from the hybrid or split radius cutting edge 22; in some implementations, the outwardly-projecting angle θ_(W)′ is approximately equal to 25′. The substantially linear, flat or planar laser beam functional entrance face segment 12 a is configured to refract the laser beam L at the laser beam entrance face 12 according to Snell's law.

As seen at FIG. 26, in some instances, the substantially linear, flat or planar laser beam functional entrance face segment 12 a of the protruding or outwardly-projecting partial wedge shape of the laser beam entrance face 12 of the laser-transmitting machining tool 10 o receives the laser beam entrance segment L_(E1) of the laser beam L and thereafter causes the laser beam refracted segment L₁ of the laser beam L to remain collimated after the laser beam L enters the body of the laser-transmitting machining tool 10 o. As can readily be seen in FIG. 26, the laser beam L is refracted by the entrance face 12 according to Snell's law, the refracted laser beam assuming an angle θ_(R) relative to a line R normal to the entrance face 12. θ_(R) may be expressed by the following equation:

$\begin{matrix} {\theta_{R} = {\sin^{- 1}\left( \frac{\sin \mspace{11mu} \theta_{W}^{\prime}}{n_{2}} \right)}} & (55) \end{matrix}$

Referring to FIG. 26′, the laser beam refracted segment L₁ of the laser beam L may be incident upon the hybrid or split radius cutting edge 22. In some instances, the laser beam refracted segment L₁ of the laser beam L is incident upon the second portion 21 d ₂ of the second downstream sidewall surface or face 21 b that defines the non-linear, curved, or arcuate portion of the hybrid or split radius cutting edge 22. Thereafter, the laser beam refracted segment L₁ of the laser beam L is refracted off the second portion 21 d ₂ of the second downstream sidewall surface or face 21 b that defines the non-linear, curved, or arcuate portion of the hybrid or split radius cutting edge 222 to define the laser beam exit segment L_(E2). After the laser beam refracted segment L₁ contacts and travels through the hybrid or split radius cutting edge 22, the laser beam exit segment L_(E2) of the laser beam L becomes convergent.

FIG. 32 is schematic view of an example computing device 3200 that may be used to implement the systems and methods described in this document (e.g., the computing device 3200 may be connected to or be a component of a beam profiler (not shown) having beam characterization software for generating the laser beam L that is received by any of the exemplary laser-transmitting machining tools 10 a-10 o described above). The computing device 3200 is intended to represent various forms of digital computers, such as laptops, desktops, workstations, personal digital assistants, servers, blade servers, mainframes, and other appropriate computers. The components shown here, their connections and relationships, and their functions, are meant to be exemplary only, and are not meant to limit implementations of the inventions described and/or claimed in this document.

The computing device 3200 includes a processor 3210 (also referred to as data processing hardware), memory 3220 (also referred to as memory hardware), a storage device 3230, a high-speed interface/controller 3240 connecting to the memory 3220 and high-speed expansion ports 3250, and a low speed interface/controller 3260 connecting to a low speed bus 3270 and a storage device 3230. Each of the components 3210, 3220, 3230, 3240, 3250, and 3260, are interconnected using various busses, and may be mounted on a common motherboard or in other manners as appropriate. The processor 3210 can process instructions for execution within the computing device 3200, including instructions stored in the memory 3220 or on the storage device 3230 to display graphical so information for a graphical user interface (GUI) on an external input/output device, such as display 3280 coupled to high speed interface 3240. In other implementations, multiple processors and/or multiple buses may be used, as appropriate, along with multiple memories and types of memory. Also, multiple computing devices 3200 may be connected, with each device providing portions of the necessary operations (e.g., as a server bank, a group of blade servers, or a multi-processor system).

The memory 3220 stores information non-transitorily within the computing device 3200. The memory 3220 may be a computer-readable medium, a volatile memory unit(s), or non-volatile memory unit(s). The non-transitory memory 3220 may be physical devices used to store programs (e.g., sequences of instructions) or data (e.g., program state information) on a temporary or permanent basis for use by the computing device 3200. Examples of non-volatile memory include, but are not limited to, flash memory and read-only memory (ROM)/programmable read-only memory (PROM)/erasable programmable read-only memory (EPROM)/electronically erasable programmable read-only memory (EEPROM)(e.g., typically used for firmware, such as boot programs). Examples of volatile memory include, but are not limited to, random access memory (RAM), dynamic random access memory (DRAM) static random access memory (SRAM), phase change memory (PCM) as well as disks or tapes.

The storage device 3230 is capable of providing mass storage for the computing device 3200. In some implementations, the storage device 3230 is a computer-readable medium. In various different implementations, the storage device 3230 may be a floppy disk device, a hard disk device, an optical disk device, or a tape device, a flash memory or other similar solid-state memory device, or an array of devices, including devices in a storage area network or other configurations. In additional implementations, a computer program product is tangibly embodied in an information carrier. The computer program product contains instructions that, when executed, perform one or more methods, such as those described above. The information carrier is a computer- or machine-readable medium, such as the memory 3220, the storage device 3230, or memory on processor 3210.

The high-speed controller 3240 manages bandwidth-intensive operations for the computing device 3200, while the low speed controller 3260 manages lower so bandwidth-intensive operations. Such allocation of duties is exemplary only. In some implementations, the high-speed controller 3240 is coupled to the memory 3220, the display 3280 (e.g., through a graphics processor or accelerator), and to the high-speed expansion ports 3250, which may accept various expansion cards (not shown). In some implementations, the low-speed controller 3260 is coupled to the storage device 3230 and a low-speed expansion port 3290. The low-speed expansion port 3290, which may include various communication ports (e.g., USB, Bluetooth, Ethernet, wireless Ethernet), may be coupled to one or more input/output devices, such as a keyboard, a pointing device, a scanner, or a networking device such as a switch or router, e.g., through a network adapter.

The computing device 3200 may be implemented in a number of different forms, as shown in the figure. For example, it may be implemented as a standard server 3200 a or multiple times in a group of such servers 3200 a, as a laptop computer 3200 b, or as part of a rack server system 3200 c.

Various implementations of the systems and techniques described herein can be realized in digital electronic and/or optical circuitry, integrated circuitry, specially designed ASICs (application specific integrated circuits), computer hardware, firmware, software, and/or combinations thereof. These various implementations can include implementation in one or more computer programs that are executable and/or interpretable on a programmable system including at least one programmable processor, which may be special or general purpose, coupled to receive data and instructions from, and to transmit data and instructions to, a storage system, at least one input device, and at least one output device.

These computer programs (also known as programs, software, software applications or code) include machine instructions for a programmable processor, and can be implemented in a high-level procedural and/or object-oriented programming language, and/or in assembly/machine language. As used herein, the terms “machine-readable medium” and “computer-readable medium” refer to any computer program product, non-transitory computer readable medium, apparatus and/or device (e.g., magnetic discs, optical disks, memory, Programmable Logic Devices (PLDs) used to provide machine instructions and/or data to a programmable processor, including a machine-readable medium that receives machine instructions as a machine-readable signal. The term “machine-readable signal” refers to any signal used to provide machine instructions and/or data to a programmable processor.

The processes and logic flows described in this specification can be performed by one or more programmable processors executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit). Processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors, and any one or more processors of any kind of digital computer. Generally, a processor will receive instructions and data from a read only memory or a random access memory or both. The essential elements of a computer are a processor for performing instructions and one or more memory devices for storing instructions and data. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto optical disks, or optical disks. However, a computer need not have such devices. Computer readable media suitable for storing computer program instructions and data include all forms of non-volatile memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto optical disks; and CD ROM and DVD-ROM disks. The processor and the memory can be supplemented by, or incorporated in, special purpose logic circuitry.

To provide for interaction with a user, one or more aspects of the disclosure can be implemented on a computer having a display device, e.g., a CRT (cathode ray tube), LCD (liquid crystal display) monitor, or touch screen for displaying information to the user and optionally a keyboard and a pointing device, e.g., a mouse or a trackball, by which the user can provide input to the computer. Other kinds of devices can be used to provide interaction with a user as well; for example, feedback provided to the user can be any form of sensory feedback. e.g., visual feedback, auditory feedback, or tactile feedback, and input from the user can be received in any form, including acoustic, speech, or tactile input. In addition, a computer can interact with a user by sending documents to and receiving documents from a device that is used by the user; for example, by sending web pages to a web browser on a user's client device in response to requests received from the web browser.

A number of implementations have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the disclosure. Accordingly, other implementations are within the scope of the following claims. 

What is claimed is:
 1. An optomechanical tool for machining a workpiece, the optomechanical tool comprising: a body of material having an entrance face, a rake face, a flank face, a rake side face, and a flank side face, the rake side face and the flank side face are connected to the entrance face, the rake side face is connected to the rake face, the flank side face is connected to the flank face, the rake face is connected to the flank face to define an at least partially curved cutting edge, wherein the entrance face extends away from the flank side face to define a back-relief angle, wherein the rake face extends away from the rake side face to define a rake angle, and wherein the entrance face is configured to direct a light beam toward one or more of the rake face, the flank face, the rake side face, the flank side face, and the at least partially curved cutting edge and through one or more of the rake face, the flank face, and the at least partially curved cutting edge, causing the light beam to refract onto the workpiece.
 2. The optomechanical tool of claim 1, wherein the rake angle is as a negative rake angle.
 3. The optomechanical tool of claim 1, wherein the back-relief angle is an obtuse.
 4. The optomechanical tool of claim 1, wherein the back-relief angle is acute.
 5. The optomechanical tool of claim 1, wherein the back-relief angle is perpendicular.
 6. The optomechanical tool of claim 1, wherein the entrance face is substantially linear, flat or planar.
 7. The optomechanical tool of claim 1, wherein the entrance face is defined by an inwardly-projecting axial cylindrical configuration.
 8. The optomechanical tool of claim 1, wherein the entrance face is defined by an outwardly-projecting axial cylindrical configuration.
 9. The optomechanical tool of claim 1, wherein the entrance face is defined by an inwardly-projecting lateral cylindrical configuration.
 10. The optomechanical tool of claim 1, wherein the entrance face is defined by an outwardly-projecting lateral cylindrical configuration.
 11. The optomechanical tool of claim 1, wherein the entrance face is defined by an inwardly-projecting spherical configuration.
 12. The optomechanical tool of claim 1, wherein the entrance face is defined by an outwardly-projecting spherical configuration.
 13. The optomechanical tool of claim 1, wherein the entrance face is defined by one or more diffractive surface portions.
 14. The optomechanical tool of claim 1, wherein one or more of the entrance face, the rake face, the flank face, the rake side face, and the flank side face is at least partially coated with a reflection-enhancing coating material.
 15. The optomechanical tool of claim 14, wherein the reflection-enhancing coating material is aluminum, silver, gold, Inconel, chrome, nickel, or titanium nitride.
 16. The optomechanical tool of claim 1, wherein the entrance face further comprises a functional entrance face segment and a non-functional entrance face segment, the functional entrance face segment is connected to the rake side face, the non-functional entrance face segment is connected to the flank side face, wherein the non-functional entrance face segment extends away from the functional entrance face segment to define a back-relief angle.
 17. The optomechanical tool of claim 1, wherein the flank face extends away from the flank side face to define a clearance angle.
 18. The optomechanical tool of claim 1, further comprising a secondary clearance face, wherein the secondary clearance face extends between and connects the flank face to the flank side face, wherein the flank face extends away from the secondary clearance face to define a clearance angle, wherein the secondary clearance face extends away from the flank side face to define a secondary clearance angle.
 19. The optomechanical tool of claim 1, further comprising a first upstream sidewall surface or face and a second upstream sidewall surface or face extending from the entrance face and connecting the rake side face to the flank side face, wherein the entrance face is defined by a wedge shape including a first light beam entrance face segment and a second light beam entrance face segment that meet at an entrance face edge, the first light beam entrance face segment is connected to the first upstream sidewall surface or face, the second light beam entrance face segment is connected to the second upstream sidewall surface or face.
 20. The optomechanical tool of claim 19, wherein the wedge shape is defined by a recessed, inverted or inwardly-projecting configuration.
 21. The optomechanical tool of claim 19, wherein the wedge shape is defined by a protruding or outwardly-projecting configuration.
 22. The optomechanical tool of claim 1, wherein the entrance face defines an optical lens housing to configured to retain an optical lens.
 23. The optomechanical tool of claim 1, wherein the body of material comprises a material selected from the group consisting of diamonds, sapphires, moissanites, chrysoberyls, alexandrite, carbides, cubic boron nitride, silicon, nitrides, steels, alloys, ceramics, alumina, glass, and glass composites.
 24. The optomechanical tool of claim 1, wherein the body of material comprises a diamond material.
 25. The optomechanical tool of claim 1, wherein a portion of a first downstream sidewall surface or face extending along the rake face defines a linear, non-curved, or non-arcuate portion of the at least partially curved cutting edge, wherein a portion of a second downstream sidewall surface or face extending along the rake face defines a non-linear, curved, or arcuate portion of the at least partially curved cutting edge, wherein the at least partially curved cutting edge is a hybrid or split radius cutting edge.
 26. An optomechanical tool for machining a workpiece, the optomechanical tool comprising: a body of material having an entrance face, a rake face, a flank face, and a flank side face, the rake face and the flank side face are connected to the entrance face, the flank side face is connected to the flank face, the rake face is connected to the flank face to define an at least partially curved cutting edge, wherein the entrance face extends away from the flank side face to define a back-relief angle, wherein the rake face extends away from the flank face to define a rake angle, wherein the entrance face is configured to direct a light beam toward one or more of the rake face, the flank face, the flank side face, and the at least partially curved cutting edge and through one or more of the rake face, the flank face, and the at least partially curved cutting edge, causing the light beam to refract onto the workpiece.
 27. The optomechanical tool of claim 26, wherein the rake angle is as a positive rake angle.
 28. The optomechanical tool of claim 26, wherein the back-relief angle is perpendicular.
 29. The optomechanical tool of claim 26, wherein the entrance face is substantially linear, flat or planar.
 30. The optomechanical tool of claim 26, further comprising a secondary clearance face, wherein the secondary clearance face extends between and connects the flank face to the flank side face, wherein the flank face extends away from the secondary clearance face to define a clearance angle, wherein the secondary clearance face extends away from the flank side face to define a secondary clearance angle.
 31. The optomechanical tool of claim 26, wherein the body of material comprises a material selected from the group consisting of diamonds, sapphires, moissanites, chrysoberyls, alexandrite, carbides, cubic boron nitride, silicon, nitrides, steels, alloys, ceramics, alumina, glass, and glass composites.
 32. The optomechanical tool of claim 26, wherein the body of material comprises a diamond material.
 33. A system comprising: an optomechanical tool including a body of material having an entrance face, a rake face, a flank face, a rake side face, and a flank side face, the rake side face and the flank side face are connected to the entrance face, the rake side face is connected to the rake face, the flank side face is connected to the flank face, the rake face is connected to the flank face to define an at least partially curved cutting edge, wherein the entrance face extends away from the flank side face to define a back-relief angle, wherein the rake face extends away from the rake side face to define a rake angle, wherein the entrance face is configured to direct a light beam toward one or more of the rake face, the flank face, the rake side face, the flank side face, and the at least partially curved cutting edge and through one or more of the rake face, the flank face, and the at least partially curved cutting edge, causing the light beam to refract onto a workpiece; and an optical lens system arranged upstream of the entrance face of the optomechanical tool.
 34. The system of claim 33, wherein the optical lens system includes: an optical lens, and a movement actuator connected to the optical lens that is configured to laterally-shift the optical lens relative to the entrance face of the optomechanical tool.
 35. The system of claim 33, wherein the optomechanical tool includes a first upstream sidewall surface or face and a second upstream sidewall surface or face extending from the entrance face and connecting the rake side face to the flank side face, wherein the entrance face is defined by a third order polynomial surface having a non-linear, arcuate, or curved configuration extending between the first upstream sidewall surface or face and the second upstream sidewall surface or face.
 36. The system of claim 33, wherein a downstream side of the optical lens is defined by a third order polynomial surface having a non-linear, arcuate, or curved configuration that extends between a first end and a second end of the optical lens.
 37. The system of claim 33, wherein the body of material of the optomechanical tool and the optical lens comprises a material selected from the group consisting of diamonds, sapphires, moissanites, chrysoberyls, alexandrite, carbides, cubic boron nitride, silicon, nitrides, steels, alloys, ceramics, alumina, glass, and glass composites.
 38. The system of claim 33, wherein the body of material of the optomechanical tool and the optical lens comprises a diamond material.
 39. A system comprising: an optomechanical tool including a body of material having an entrance face, a rake face, a flank face, a rake side face, and a flank side face, the rake side face and the flank side face are connected to the entrance face, the rake side face is connected to the rake face, the flank side face is connected to the flank face, the rake face is connected to the flank face to define an at least partially curved cutting edge, wherein the entrance face extends away from the flank side face to define a back-relief angle, wherein the rake face extends away from the rake side face to define a rake angle, wherein the entrance face is configured to direct a light beam toward one or more of the rake face, the flank face, the rake side face, the flank side face, and the at least partially curved cutting edge and through one or more of the rake face, the flank face, and the at least partially curved cutting edge, causing the light beam to refract onto a workpiece; and an optical prism system arranged upstream of the entrance face of the optomechanical tool.
 40. The system of claim 39, wherein the optical prism system includes: a first right angle prism; a second right angle prism; and a movement actuator connected to the second right angle prism that is configured to axially-shift the second right angle prism relative to the entrance face of the optomechanical tool and the first right angle prism.
 41. The system of claim 40, wherein the body of material of the optomechanical tool, the first right angle prism, and the second right angle prism comprises a material selected from the group consisting of diamonds, sapphires, moissanites, chrysoberyls, alexandrite, carbides, cubic boron nitride, silicon, nitrides, steels, alloys, ceramics, alumina, glass, and glass composites.
 42. The system of claim 40, wherein the body of material of the optomechanical tool, the first right angle prism, and the second right angle prism comprises a diamond material.
 43. A method of a light beam toward a workpiece comprising: providing an optomechanical tool defined by an entrance face, a rake face, a flank face connected to the rake face, a rake side face extending between the entrance face and the rake face, and a flank side face extending between the entrance face and the flank face, wherein the connection of the rake face to the flank face defines an at least partially curved cutting edge; receiving the light beam at the entrance face; refracting the light beam by toward a reflecting face defined by one or more of the rake side face and the flank side face, reflecting the light beam toward one or more of the rake face, the flank face, and the at least partially curved cutting edge; and refracting the light beam through the rake face, the flank face, and the at least partially curved cutting edge toward the workpiece.
 44. The method of claim 43, wherein the reflecting face is the flank side face.
 45. The method of claim 43, wherein the optomechanical tool further comprises a secondary clearance face extending between the flank face and the flank side face, wherein the secondary clearance face extends between the flank face and the flank side face to define a secondary clearance angle, wherein the reflecting face further defined by the secondary clearance face.
 46. The method of claim 45, wherein the secondary clearance angle is between 120° and 180°.
 47. The method of claim 43, wherein the optomechanical tool is formed from a material selected from the group consisting of diamonds, sapphires, moissanites, chrysoberyls, alexandrite, carbides, cubic boron nitride, silicon, nitrides, steels, alloys, ceramics, alumina, glass, and glass composites.
 48. The method of claim 43, wherein the reflecting face is entirely coated or partially coated with a reflection-enhancing coating material.
 49. The method of claim 48, wherein the reflection-enhancing coating material is aluminum, silver, gold, Inconel, chrome, nickel, or titanium nitride.
 50. The method of claim 43, wherein the entrance face is defined by a curved surface.
 51. The method of claim 43, wherein the light beam is defined by an initial focal point or focal plane, wherein the entrance face is configured to refract the light beam to define a transformed focal point or focal plane. 